A quantity with an exponent has three components--the base, the
exponent, and the coefficient.
Example:In the quantity 3x^5, the coefficient is 3, the base is x, and the exponent is 5.
**note x^3 is x raised to power 3
Quotient Law :a ^m / a ^n = a^ (m – n) where m, n are real terms and a not equal to 0
In division, If base of two terms is same then subtract the exponent of denominator from the exponent of numerator
*This rule is applicable only when base is same8
Example:simplify
7 ^6 ÷ 7^ 4
7^ (6-4) (here Base is same,using quotient rule)
7 ^2
Practice:simplify
5 ^8 / 5 ^5
7^6 ÷ 7^5
2^5 ÷ 2^9
Answers:
5^3
7^1
2^(-4)
Example:simplify
8^6 ÷ 8^ -4
8^ (6-(-4)) (here Base is same,using quotient rule)
8^(6 +4)
8^10
Practice:simplify
3^4 ÷ 3^-5
12^5 ÷ 12^-5
5^7 ÷ 5-3
Answers:
3^9
12^10
5^10
Practice:simplify
x^8 ÷ x^-3
y^5 ÷ y^3
p^7 ÷ p^4
Answers:
x^11
y^2
p^3
Wednesday, August 20, 2008
Product Law for Exponents:
Exponential notation or Power notation :
In a ^m , a is base m is called index or exponent or power
note : x^2 is x raised to power 2
Product Law: In multiplication , if base is same then add exponents.
a ^m * a ^n = a^ (m + n) where m, n are real terms and a not equal to zero
*This rule is applicable only when base is same
Example :simplify 2^ 5 * 2^ 6
2^ (5 + 6) = 2^11 ( using product law,here base is same )
practice : simplify
3^ 5 * 3^ 8
4^ 5 * 4 ^6
9^ 5 * 9 ^ 12
Answers:
3^13
4^11
9^17
Example :simplify x^ 6 * x^ 7
x^ (6 + 7) = x^13 ( using product law,here base is same )
practice: simplify
X * X^3 * X ^5 (note : x = x^1)
A^2 * A^4 * A^3 * A
n^2 * n^9 * n^23 * n^7
Answers:
X^9
A^10
n^41
Example : simplify x * y * y^ 2 * x^ 3
x * x^3 * y * y ^2 (first arrange the letters)
x^4 * Y^3 ( using product rule)
Practice: simplify
x^2 * y^2 * x^5 * y^6
x * y * x^4 * x^3 * y^5
k * y^2 * k^3 * k^2 * y^5
Answers:
x^7 * y^8
x^8 * y^6
k^6 * y 7
In a ^m , a is base m is called index or exponent or power
note : x^2 is x raised to power 2
Product Law: In multiplication , if base is same then add exponents.
a ^m * a ^n = a^ (m + n) where m, n are real terms and a not equal to zero
*This rule is applicable only when base is same
Example :simplify 2^ 5 * 2^ 6
2^ (5 + 6) = 2^11 ( using product law,here base is same )
practice : simplify
3^ 5 * 3^ 8
4^ 5 * 4 ^6
9^ 5 * 9 ^ 12
Answers:
3^13
4^11
9^17
Example :simplify x^ 6 * x^ 7
x^ (6 + 7) = x^13 ( using product law,here base is same )
practice: simplify
X * X^3 * X ^5 (note : x = x^1)
A^2 * A^4 * A^3 * A
n^2 * n^9 * n^23 * n^7
Answers:
X^9
A^10
n^41
Example : simplify x * y * y^ 2 * x^ 3
x * x^3 * y * y ^2 (first arrange the letters)
x^4 * Y^3 ( using product rule)
Practice: simplify
x^2 * y^2 * x^5 * y^6
x * y * x^4 * x^3 * y^5
k * y^2 * k^3 * k^2 * y^5
Answers:
x^7 * y^8
x^8 * y^6
k^6 * y 7
Exponents
Exponential notation or Power notation :
In a ^m , a is base m is called index or exponent or power
note : x^2 is x raised to power 2
Example : 5 ^4 = 5 * 5 * 5 * 5 ( 5 raised to power four)
a ^3 = a * a * a
x^ 7 = x * x * x * x * x * x * x
2 ^5 = 2 * 2 * 2 * 2 * 2
3^ 9 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3
practice:write expansion of
5 ^5 =
2^ 8 =
a ^6 =
x ^4 =
Example: find the value of 2^4
2 ^4 = 2 * 2 * 2 * 2 = 16
practice :find the value of
5^ 3
4^ 4
8^ 2
Answers:
125
256
64
Example :Express 81 as a power of 3 ( in power notation)
81 = 3 * 3 * 3 * 3 = 3 ^4 ( 3 raised to power 4)
practice : Express the following in power notation
64 as a power of 4
49 as a power of 7
625 as a power of 5
Answers:
4^3
7^2
5^4
Example : Simplify 3^4 + 5^3
(3 *3 * 3 * 3 ) + ( 5 * 5 * 5 )
81 + 125
206 is the answer
practice: simplify
5^6 + 7 ^3
9^3 + 6^4
2^6 + 3^5
Answers:
15968
2024
307
Example : simplify 2^5 - 5^2
(2 * 2 * 2 * 2 * 2 ) - ( 5 * 5 )
32 - 25 = 7 is the answer
practice : simplify
4^5 - 3^4
3^6 - 6^3
8^3 + 5^3
Answers:
943
513
6377^2 - 2^5
In a ^m , a is base m is called index or exponent or power
note : x^2 is x raised to power 2
Example : 5 ^4 = 5 * 5 * 5 * 5 ( 5 raised to power four)
a ^3 = a * a * a
x^ 7 = x * x * x * x * x * x * x
2 ^5 = 2 * 2 * 2 * 2 * 2
3^ 9 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3
practice:write expansion of
5 ^5 =
2^ 8 =
a ^6 =
x ^4 =
Example: find the value of 2^4
2 ^4 = 2 * 2 * 2 * 2 = 16
practice :find the value of
5^ 3
4^ 4
8^ 2
Answers:
125
256
64
Example :Express 81 as a power of 3 ( in power notation)
81 = 3 * 3 * 3 * 3 = 3 ^4 ( 3 raised to power 4)
practice : Express the following in power notation
64 as a power of 4
49 as a power of 7
625 as a power of 5
Answers:
4^3
7^2
5^4
Example : Simplify 3^4 + 5^3
(3 *3 * 3 * 3 ) + ( 5 * 5 * 5 )
81 + 125
206 is the answer
practice: simplify
5^6 + 7 ^3
9^3 + 6^4
2^6 + 3^5
Answers:
15968
2024
307
Example : simplify 2^5 - 5^2
(2 * 2 * 2 * 2 * 2 ) - ( 5 * 5 )
32 - 25 = 7 is the answer
practice : simplify
4^5 - 3^4
3^6 - 6^3
8^3 + 5^3
Answers:
943
513
6377^2 - 2^5
Laws of exponents - power law
A quantity with an exponent has three components--the base, the
exponent, and the coefficient.
**note x^9 is x raised to power 9**
In the quantity 3x^5, the coefficient is 3, the base is x, and the exponent is 5.
Power law :(a ^m) ^n = a ^ m n (the new exponent is product of two exponents)
Example: simplify (5^ 7)^6
5^ (7 * 6)=5^42
practice: simplify
( 2^ 5 )^ 8
( 3^ 6 ) ^5
( 7^ 8 )^ 6
Answers:
2^40
3^30
7^48
Example: simplify (x^ 3)^4
x^ (3* 4)
x^12
practice: simplify
( p^ 5 )^ 3
( y^ 7 ) ^4
( a^ 9 )^ 5
Answers:
p^15
y^28
a^45
exponent, and the coefficient.
**note x^9 is x raised to power 9**
In the quantity 3x^5, the coefficient is 3, the base is x, and the exponent is 5.
Power law :(a ^m) ^n = a ^ m n (the new exponent is product of two exponents)
Example: simplify (5^ 7)^6
5^ (7 * 6)=5^42
practice: simplify
( 2^ 5 )^ 8
( 3^ 6 ) ^5
( 7^ 8 )^ 6
Answers:
2^40
3^30
7^48
Example: simplify (x^ 3)^4
x^ (3* 4)
x^12
practice: simplify
( p^ 5 )^ 3
( y^ 7 ) ^4
( a^ 9 )^ 5
Answers:
p^15
y^28
a^45
Exponents - zero and negative exponents
Exponential notation :
A quantity with an exponent has three components--the base, the
exponent, and the coefficient.
In the quantity 3x^5, the coefficient is 3, the base is x, and the exponent is 5.
In the quantity 3(16)^7x, the coefficient is 3, the base is 16, and the exponent is 7x.
In the quantity 26(2y)^xy, the coefficient is 16, the base is 2y, and the exponent is xy.
In the quantity r^2, the (implied) coefficient is 1, the base is r, and the exponent is 2.
In the quantity 2y, the coefficient is 1, the base is y, and the (implied) exponent is 1.
**note x^9 is x raised to power 9 **
Negative Exponents
Taking a number to a negative exponent does not necessarily yield a negative answer.
The number raised to a negative exponent is a rational number.
a ^ -m = 1/a^m( the exponents becomes positive when it comes to denominator)
Example: find
(2) ^– 2 = 1 /2^2 =1/4
practice:find
7^-3
4^-2
6^-4
Answers:
1/343
1/16
1/1296
***If the base number is a fraction, then the negative exponent switches the numerator and the denominator***.
Example: find(2/3)^ -4
(3/2)^ 4
(3^ 4)/(2^ 4)
81/16
Example :find (- 5/6)^ -3
(6/(- 5)) ^3
(6 ^3)/((- 5) ^3)
216/(- 125)
- 216/125.
Practice : find
(7/5) ^-2
(-3/11)^ -1
(6/(-7)) ^-3
Answers:
25/49
-11/3
-343/216
Zero power or exponent
Any number raised to the zero power as 1
a^0 = 1
Example:
8^0 = 1
(- 17)^0 = 1
practice: find
(-6)^0
234^0
12^0
x^0
(1/8)^0
8.2365^0
Answers :
1
1
1
1
1
1
Laws of Exponents :
a^ 1 = a (If any variable or constant is written without an exponent , it is understood to have an exponent of 1)
a^ m * a ^n = a^( m + n) ( product Law for Exponents)
a ^m / a^ n = a ^(m – n) ( quotient Law for Exponents)
(a ^m) ^n = a ^ m n (power law for exponents)
A quantity with an exponent has three components--the base, the
exponent, and the coefficient.
In the quantity 3x^5, the coefficient is 3, the base is x, and the exponent is 5.
In the quantity 3(16)^7x, the coefficient is 3, the base is 16, and the exponent is 7x.
In the quantity 26(2y)^xy, the coefficient is 16, the base is 2y, and the exponent is xy.
In the quantity r^2, the (implied) coefficient is 1, the base is r, and the exponent is 2.
In the quantity 2y, the coefficient is 1, the base is y, and the (implied) exponent is 1.
**note x^9 is x raised to power 9 **
Negative Exponents
Taking a number to a negative exponent does not necessarily yield a negative answer.
The number raised to a negative exponent is a rational number.
a ^ -m = 1/a^m( the exponents becomes positive when it comes to denominator)
Example: find
(2) ^– 2 = 1 /2^2 =1/4
practice:find
7^-3
4^-2
6^-4
Answers:
1/343
1/16
1/1296
***If the base number is a fraction, then the negative exponent switches the numerator and the denominator***.
Example: find(2/3)^ -4
(3/2)^ 4
(3^ 4)/(2^ 4)
81/16
Example :find (- 5/6)^ -3
(6/(- 5)) ^3
(6 ^3)/((- 5) ^3)
216/(- 125)
- 216/125.
Practice : find
(7/5) ^-2
(-3/11)^ -1
(6/(-7)) ^-3
Answers:
25/49
-11/3
-343/216
Zero power or exponent
Any number raised to the zero power as 1
a^0 = 1
Example:
8^0 = 1
(- 17)^0 = 1
practice: find
(-6)^0
234^0
12^0
x^0
(1/8)^0
8.2365^0
Answers :
1
1
1
1
1
1
Laws of Exponents :
a^ 1 = a (If any variable or constant is written without an exponent , it is understood to have an exponent of 1)
a^ m * a ^n = a^( m + n) ( product Law for Exponents)
a ^m / a^ n = a ^(m – n) ( quotient Law for Exponents)
(a ^m) ^n = a ^ m n (power law for exponents)
slope or gradient of line passing through two points
slope of line passing through two points:
For the points ( x1,y1) and (x2 ,y2)
slope =y2 - y1/x2 -x1 (Formula)
It means ...
Slope = Difference of y coordinates /Difference of x coordinates
Example :find the slope or gradient of line passing through (3,-2) and (-1,4)
slope =y2 - y1/x2 -x1
slope = 4-(-2) / -1-3 ( x1 = 3,y1 = -2 ,x2 = -1 , y2 = 4, using slope formula)
slope = 4 + 2 / - 4
slope = 6 / -4
slope = -3/2
Practice:
find the slope or gradient of line passing through (2,3) and (4,6)
find the slope or gradient of line passing through (0,-3) and (2,1)
find the slope or gradient of line passing through (1,-2) and (2,-3)
Answers:
3/2
-2
-1
Writing equation from graph
To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope.
Slope = Difference of y coordinates /Difference of x coordinates
For the points ( x1,y1) and (x2 ,y2)
slope = y2 - y1/x2 -x1
This is the value of M in the equation. Next, find the coordinates of the y-intercept--this should be of the form (0, b). The y- coordinate is the value of b in the equation.
Finally, write the equation, substituting numerical values in for M and b.
Y = MX + b
For the points ( x1,y1) and (x2 ,y2)
slope =y2 - y1/x2 -x1 (Formula)
It means ...
Slope = Difference of y coordinates /Difference of x coordinates
Example :find the slope or gradient of line passing through (3,-2) and (-1,4)
slope =y2 - y1/x2 -x1
slope = 4-(-2) / -1-3 ( x1 = 3,y1 = -2 ,x2 = -1 , y2 = 4, using slope formula)
slope = 4 + 2 / - 4
slope = 6 / -4
slope = -3/2
Practice:
find the slope or gradient of line passing through (2,3) and (4,6)
find the slope or gradient of line passing through (0,-3) and (2,1)
find the slope or gradient of line passing through (1,-2) and (2,-3)
Answers:
3/2
-2
-1
Writing equation from graph
To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope.
Slope = Difference of y coordinates /Difference of x coordinates
For the points ( x1,y1) and (x2 ,y2)
slope = y2 - y1/x2 -x1
This is the value of M in the equation. Next, find the coordinates of the y-intercept--this should be of the form (0, b). The y- coordinate is the value of b in the equation.
Finally, write the equation, substituting numerical values in for M and b.
Y = MX + b
Saturday, August 16, 2008
Linear Equation (Point slope form)
When we know the slope and one point which is not the y-intercept, we can write the equation in point-slope form.
Equations in point-slope form look like this:
Y - k = M(X - h)
where M is the slope of the line and (h, k) is a point on the line (any point works).
Example : Write an equation of the line which passes through (3, 4) and has slope m = 5
h = 3 and k = 4.
Y - k = M(X - h) point-slope form
y - 4 = 5(x - 3) (use distributive property a(b+c) = a*b+a*c, 5 ( x - 3 ) = 5 x - 15)
Y – 4 = 5x – 15 (adding 4 to both sides)
Y = 5x -11
Practice:
Write an equation of the line which passes through (- 3, - 7) and has slope m = 3
Write an equation of the line which passes through (- 1, - 1) and has slope m = 2
Write an equation of the line which passes through (1, 2) and has slope m = -3
Answers:
y = 3x + 8
y = 2x + 1
y = -3x + 5
Equations in point-slope form look like this:
Y - k = M(X - h)
where M is the slope of the line and (h, k) is a point on the line (any point works).
Example : Write an equation of the line which passes through (3, 4) and has slope m = 5
h = 3 and k = 4.
Y - k = M(X - h) point-slope form
y - 4 = 5(x - 3) (use distributive property a(b+c) = a*b+a*c, 5 ( x - 3 ) = 5 x - 15)
Y – 4 = 5x – 15 (adding 4 to both sides)
Y = 5x -11
Practice:
Write an equation of the line which passes through (- 3, - 7) and has slope m = 3
Write an equation of the line which passes through (- 1, - 1) and has slope m = 2
Write an equation of the line which passes through (1, 2) and has slope m = -3
Answers:
y = 3x + 8
y = 2x + 1
y = -3x + 5
Linear equations ( slope -intercept form)
The first of the forms for a linear equation is slope-intercept form.
Y = MX + b is slope-intercept form
where M is the slope or gradient of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line crosses the y-axis.
Example : In y = 2x + 9 ,slope 2 and y intercept is 9.
Practice: find slope and y intercept of the equations
y = 5x + 12
y = 8x + 5
y = 7x + 4
Answers:
slope = 5 and y intercept = 12
slope = 8 and y intercept = 5
slope = 7 and y intercept = 4
Example: write the equation of line whose slope is 5 and y intercept is 2
y = 5x + 2
practice: write the equation of line
slope = 4 & y intercept =7
slope = 2 & y intercept =9
slope = 5 & y intercept = -12
Answers:
y =4x + 7
y =2x + 9
y =5x - 12
Example : Write an equation of the line with slope m = 6 which crosses the y-axis at (0, 5). (here y intercept is 5, because x coordinates is 0)
y = 6x + 5
practice:
Write an equation of the line with slope m = 3 which crosses the y-axis at (0, 4).
Write an equation of the line with slope m = -5 which crosses the y-axis at (0, 9).
Write an equation of the line with slope m = 1/2 which crosses the y-axis at (0, 3/2).
Answers:
y = 3x + 4
y = -5x + 9
2y = x =3
Y = MX + b is slope-intercept form
where M is the slope or gradient of the line and b is the y-intercept of the line, or the y-coordinate of the point at which the line crosses the y-axis.
Example : In y = 2x + 9 ,slope 2 and y intercept is 9.
Practice: find slope and y intercept of the equations
y = 5x + 12
y = 8x + 5
y = 7x + 4
Answers:
slope = 5 and y intercept = 12
slope = 8 and y intercept = 5
slope = 7 and y intercept = 4
Example: write the equation of line whose slope is 5 and y intercept is 2
y = 5x + 2
practice: write the equation of line
slope = 4 & y intercept =7
slope = 2 & y intercept =9
slope = 5 & y intercept = -12
Answers:
y =4x + 7
y =2x + 9
y =5x - 12
Example : Write an equation of the line with slope m = 6 which crosses the y-axis at (0, 5). (here y intercept is 5, because x coordinates is 0)
y = 6x + 5
practice:
Write an equation of the line with slope m = 3 which crosses the y-axis at (0, 4).
Write an equation of the line with slope m = -5 which crosses the y-axis at (0, 9).
Write an equation of the line with slope m = 1/2 which crosses the y-axis at (0, 3/2).
Answers:
y = 3x + 4
y = -5x + 9
2y = x =3
Problems on Simple Simultaneous Equations
Problems related with numbers.
word problems appear confusing, and it is difficult to know where to begin.Read the problem.Form the equation, then solve it.
Example: If one number is thrice the other and their sum is 40, find the numbers.
Let the numbers be x and y.
x is 3 times y
x = 3y , Equation(1)
Sum of x and y is 40
x + y = 40 , Equation (2)
Putting the value of x from (1) in (2), we get,
3y + y = 40
4y/4 = 40/4 (dividing both sides by 4)
y = 10
Substituting y = 10 in (1), we get,
x = 3 x 10
x= 30
The required numbers are 10 and 30.
Practice :
The sum of 2 numbers is 50 and their difference is 16.find the numbers
The sum of two numbers is 43.if the larger is doubled and the smaller is tripled, the difference is 36. find the two numbers.
Example:
Six years hence a man's age will be three times his son's age, and three years ago he was nine times as old as his son. Find their present ages.
Let the present age of the man be x years, and the present age of his son be y years.
6 years hence their ages will be (x + 6) years and (y + 6) years.
x + 6 = 3(y + 6)
x + 6 = 3y + 18
x - 3y = 18 - 6
x - 3y = 12 Equation(1)
3 years ago, their ages were (x - 3) years and (y - 3) years.
x - 3 = 9(y - 3)
x - 3 = 9y – 27(adding 3 to both sides)
x = 9y – 24
x – 9y = -24 equation (2)
Subtracting (1) from (2) -6y = -36(dividing both sides by -6)
-6y/-6 =-36/-6
y = 6
Substituting y = 6 in (1), we get
x - 3*6 = 12
x - 18 = 12
x = 12 + 18 = 30
The present age of the man is 30 years and the present age of his son is 6 years.
Practice:
The present age of the father is four times that of his son. Six years hence the age of the father will be thrice that of his son. Find their present ages.
The age of a man is three times the age of his daughter and five years hence his age will be double of his Son's age.find their present age
word problems appear confusing, and it is difficult to know where to begin.Read the problem.Form the equation, then solve it.
Example: If one number is thrice the other and their sum is 40, find the numbers.
Let the numbers be x and y.
x is 3 times y
x = 3y , Equation(1)
Sum of x and y is 40
x + y = 40 , Equation (2)
Putting the value of x from (1) in (2), we get,
3y + y = 40
4y/4 = 40/4 (dividing both sides by 4)
y = 10
Substituting y = 10 in (1), we get,
x = 3 x 10
x= 30
The required numbers are 10 and 30.
Practice :
The sum of 2 numbers is 50 and their difference is 16.find the numbers
The sum of two numbers is 43.if the larger is doubled and the smaller is tripled, the difference is 36. find the two numbers.
Example:
Six years hence a man's age will be three times his son's age, and three years ago he was nine times as old as his son. Find their present ages.
Let the present age of the man be x years, and the present age of his son be y years.
6 years hence their ages will be (x + 6) years and (y + 6) years.
x + 6 = 3(y + 6)
x + 6 = 3y + 18
x - 3y = 18 - 6
x - 3y = 12 Equation(1)
3 years ago, their ages were (x - 3) years and (y - 3) years.
x - 3 = 9(y - 3)
x - 3 = 9y – 27(adding 3 to both sides)
x = 9y – 24
x – 9y = -24 equation (2)
Subtracting (1) from (2) -6y = -36(dividing both sides by -6)
-6y/-6 =-36/-6
y = 6
Substituting y = 6 in (1), we get
x - 3*6 = 12
x - 18 = 12
x = 12 + 18 = 30
The present age of the man is 30 years and the present age of his son is 6 years.
Practice:
The present age of the father is four times that of his son. Six years hence the age of the father will be thrice that of his son. Find their present ages.
The age of a man is three times the age of his daughter and five years hence his age will be double of his Son's age.find their present age
Friday, August 15, 2008
Linear Equations with one varable
An equation of the type ax + b = 0 , a is not equal to zero is called a linear equation in the variable x.
Example :solve for x , 2x + 5 = 10 - 3x
( bring x terms one side and numbers other side,when - 3x comes to left side it becoms +3x, +5 becomes - 5 when it comes to right side)
2x + 3x = 10-5
5x=5(divide both side by 5)
5x/5=5 / 5
x = 1
Practice:solve for x
8 x + 8 = 5 x + 19
5 x - 3 = 3 x - 5
4 p + 2 = 9 - 3 p
Answers:
x = 9
x = -1
p = 1
Example: Solve for x , 6x - 5 - 2x + 3 - 2 = 4
First, simplify the equation by combining like terms
(like terms are terms containing same literal example 2x , -7x ,8/7x)
4x - 4 = 4
4x - 4 + 4 = 4 + 4 ( adding 4 to both sides)
4x = 8
4x / 4 = 8 / 4 (dividing both sides by 4)
x = 2
Practice: solve for x
m + 13m - 13 = 15
7 = 8 + 5j – 9 + 3j
d + 8 - 14 + 14d = 39
Answers:
m = 2
j = 1
d = 3
Example: solve for x , ( 3 x - 1) – 2 ( x – 5 ) = 3 ( 5 – x )+ 6
(use distributive property, a(b+c) = a*b + a*c ), -2(x-5)= -2x+ 10 , 3(5-x)=15-3x
3 x – 1 - 2 x + 10 = 15 – 3 x + 6
3 x -2 x + 3 x = 15 + 6 + 1 – 10 (bring x terms to left side,numbers to right side)
4 x = 12 (combining like terms)
4x/ 4= 12/4 (dividing both side by 4)
x= 3
Practice:solve for x
2( x – 4 ) – 3 ( x + 2 ) = 4 ( x + 1 ) – 8
3( y - 7 ) - 2( 3y - 4 ) = ( 2- 5y )+ 3
7 – 3 (2x-1) = 4 (5 – x)- 7x
Answers:
x = -2
y = 2
x = 2
Example :solve for x , 2x + 5 = 10 - 3x
( bring x terms one side and numbers other side,when - 3x comes to left side it becoms +3x, +5 becomes - 5 when it comes to right side)
2x + 3x = 10-5
5x=5(divide both side by 5)
5x/5=5 / 5
x = 1
Practice:solve for x
8 x + 8 = 5 x + 19
5 x - 3 = 3 x - 5
4 p + 2 = 9 - 3 p
Answers:
x = 9
x = -1
p = 1
Example: Solve for x , 6x - 5 - 2x + 3 - 2 = 4
First, simplify the equation by combining like terms
(like terms are terms containing same literal example 2x , -7x ,8/7x)
4x - 4 = 4
4x - 4 + 4 = 4 + 4 ( adding 4 to both sides)
4x = 8
4x / 4 = 8 / 4 (dividing both sides by 4)
x = 2
Practice: solve for x
m + 13m - 13 = 15
7 = 8 + 5j – 9 + 3j
d + 8 - 14 + 14d = 39
Answers:
m = 2
j = 1
d = 3
Example: solve for x , ( 3 x - 1) – 2 ( x – 5 ) = 3 ( 5 – x )+ 6
(use distributive property, a(b+c) = a*b + a*c ), -2(x-5)= -2x+ 10 , 3(5-x)=15-3x
3 x – 1 - 2 x + 10 = 15 – 3 x + 6
3 x -2 x + 3 x = 15 + 6 + 1 – 10 (bring x terms to left side,numbers to right side)
4 x = 12 (combining like terms)
4x/ 4= 12/4 (dividing both side by 4)
x= 3
Practice:solve for x
2( x – 4 ) – 3 ( x + 2 ) = 4 ( x + 1 ) – 8
3( y - 7 ) - 2( 3y - 4 ) = ( 2- 5y )+ 3
7 – 3 (2x-1) = 4 (5 – x)- 7x
Answers:
x = -2
y = 2
x = 2
Solving long equations
For solving long equations first combine like terms, then solve the equation.
Example : solve 61 = 10 - 4 + 7 + 16 + v
61 = 10 + 7 + 16 – 4 + v
61 = 33 – 4 + v
61 = 29 + v
61 – 29 = 29 – 29 + v ( subtracting 29 from both sides)
32 = v
Practice : solve
11 + 5 + 1 + h + 7 = 83
6 + u + 8 - 17 = 17
37 = 8 + s – 15
38 = 12 + 13 + 8 + p
Example: Solve for x , 6x - 5 - 2x + 3 - 2 = 4
First, simplify the equation by combining like terms
6 x – 2x – 5 +3 - 2 = 4
4x -4 = 4 (Reverse of subtraction is addition)
4x - 4 + 4 = 4 + 4 ( adding 4 to both sides )
4x = 8 (Reverse of multiplication is division)
4x/4 = 8 /4( dividing both sides by 4)
x = 2
practice : find x
2x – 6x + 12 – 26 + 5x = 40
5p – 14 + 3p + 4p – 25 – 9p = 57
-2s + 23 + 15 -6s + 5s = -1
Example : solve 61 = 10 - 4 + 7 + 16 + v
61 = 10 + 7 + 16 – 4 + v
61 = 33 – 4 + v
61 = 29 + v
61 – 29 = 29 – 29 + v ( subtracting 29 from both sides)
32 = v
Practice : solve
11 + 5 + 1 + h + 7 = 83
6 + u + 8 - 17 = 17
37 = 8 + s – 15
38 = 12 + 13 + 8 + p
Example: Solve for x , 6x - 5 - 2x + 3 - 2 = 4
First, simplify the equation by combining like terms
6 x – 2x – 5 +3 - 2 = 4
4x -4 = 4 (Reverse of subtraction is addition)
4x - 4 + 4 = 4 + 4 ( adding 4 to both sides )
4x = 8 (Reverse of multiplication is division)
4x/4 = 8 /4( dividing both sides by 4)
x = 2
practice : find x
2x – 6x + 12 – 26 + 5x = 40
5p – 14 + 3p + 4p – 25 – 9p = 57
-2s + 23 + 15 -6s + 5s = -1
Combining like terms
Like terms are terms that contain the exact same variables raised to the same exponents.
**Note x^3 is x raised to power 3
For example, 15yz and 22yz are like terms, but 15yz^2 and 22yz are not.
12 and -6 are like terms, because they are both constant terms .
First group the coefficients of like terms together add the coefficients of
the like terms (or subtract them if they are negative).
Simplify : 11wz + 6w^2 -10wz + wz - 6 - 2w + 3w^2.
6w^2 +3w^2 + 11wz - 10wz + wz - 2w – 6
(note that this expression and the original expression both contain 7 terms)
(6 + 3)w^2 + (11 - 10 + 1)wz - 2w - 6
9w^2 + 2wz - 2w - 6
Practice : Simplify the following expressions
x + 18x^2 + 10 + 7x^2 + 9 + 13x - 14x^2
1 - 15 + 10j + 18j^2 + 12 - 8j^2 + 6j
a - 12a^2 + 11a - 4a + 13a^2 + 6a
Answers:
11x^2 +14x +19
10 j^2 +16j -2
a^2 +13a
Example : combine like terms in the expression 3x + 4y + 6x – 12y - 5x
3x + 4y + 6x – 12y - 5x ( write like terms together)
3x + 6x – 5x + 4y – 12y
9x – 5x - 8y
4x - 8y
Practice: Combine like terms in the following expressions
13z + 11z + 18x - 2y - 3z+ 6y
g + 15k - 4g - 6k + 3k
10p - 5k + 8k + 17p – 4p
Answers:
21z + 4y + 18x
-3g + 12k
23p + 3k
**Note x^3 is x raised to power 3
For example, 15yz and 22yz are like terms, but 15yz^2 and 22yz are not.
12 and -6 are like terms, because they are both constant terms .
First group the coefficients of like terms together add the coefficients of
the like terms (or subtract them if they are negative).
Simplify : 11wz + 6w^2 -10wz + wz - 6 - 2w + 3w^2.
6w^2 +3w^2 + 11wz - 10wz + wz - 2w – 6
(note that this expression and the original expression both contain 7 terms)
(6 + 3)w^2 + (11 - 10 + 1)wz - 2w - 6
9w^2 + 2wz - 2w - 6
Practice : Simplify the following expressions
x + 18x^2 + 10 + 7x^2 + 9 + 13x - 14x^2
1 - 15 + 10j + 18j^2 + 12 - 8j^2 + 6j
a - 12a^2 + 11a - 4a + 13a^2 + 6a
Answers:
11x^2 +14x +19
10 j^2 +16j -2
a^2 +13a
Example : combine like terms in the expression 3x + 4y + 6x – 12y - 5x
3x + 4y + 6x – 12y - 5x ( write like terms together)
3x + 6x – 5x + 4y – 12y
9x – 5x - 8y
4x - 8y
Practice: Combine like terms in the following expressions
13z + 11z + 18x - 2y - 3z+ 6y
g + 15k - 4g - 6k + 3k
10p - 5k + 8k + 17p – 4p
Answers:
21z + 4y + 18x
-3g + 12k
23p + 3k
Solving multi step Equations
Some equations cann't solved by single step .These equations can be solved in multi steps.
Example: Solve for x , 5x + 9 = 44
Step -1: Reverse of addition is subtraction.
5x + 9 - 9 = 44 – 9 ( subtracting 9 from both sides)
5x = 35
Step – 2 : Reverse of multiplication is division
5x/5 = 35 / 5 ( dividing both sides by 5)
x = 7
Practice :solve
2x + 3 = 7
7x + 9 = 44
22 + 3x = 41
Example: Solve for x , 2x – 7 = 1
Step -1: Reverse of subtraction is addition:
2x + 7 - 7 = 1 + 7 ( adding 7 to both sides)
2x = 8
Step – 2 :Reverse of multiplication is division :
2x /2= 8 /2( dividing both sides by 2)
x = 4
Practice :Solve
7x – 8 = 6
4y – 15 = 25
8p – 12 = 12
Example : Solve for x ,
x + 8 = 24
--
5
Step -1 :Reverse of addition is subtraction :
X + 8 – 8 = 24 – 8 ( subtracting 8 from both sides)
--
5
X = 16
--
5
Step – 2 :Reverse of division is multiplication :
X * 5 = 16 * 5 ( multiplying both sides by 5)
--
5
X = 80
Practice :Solve
P + 6 = 11
--
5
X + 24 = 35
--
6
Y + 13 = 42
--
7
Example : Solve for x
x - 14 = 5
--
3
Step -1: Reverse of subtraction is addition.
x - 14 + 14 = 5 +14 ( adding 14 to both sides)
--
3
X = 19
--
3
Step – 2 :Reverse of division is multiplication
X * 3 = 19 * 3 ( multiplying both sides by 3)
---
3
X = 57
Practice :Solve
h - 9 = 18
--
4
X
-- - 4 = 9
7
p
-- - 21 = 5
6
Example: Solve for x , 5x + 9 = 44
Step -1: Reverse of addition is subtraction.
5x + 9 - 9 = 44 – 9 ( subtracting 9 from both sides)
5x = 35
Step – 2 : Reverse of multiplication is division
5x/5 = 35 / 5 ( dividing both sides by 5)
x = 7
Practice :solve
2x + 3 = 7
7x + 9 = 44
22 + 3x = 41
Example: Solve for x , 2x – 7 = 1
Step -1: Reverse of subtraction is addition:
2x + 7 - 7 = 1 + 7 ( adding 7 to both sides)
2x = 8
Step – 2 :Reverse of multiplication is division :
2x /2= 8 /2( dividing both sides by 2)
x = 4
Practice :Solve
7x – 8 = 6
4y – 15 = 25
8p – 12 = 12
Example : Solve for x ,
x + 8 = 24
--
5
Step -1 :Reverse of addition is subtraction :
X + 8 – 8 = 24 – 8 ( subtracting 8 from both sides)
--
5
X = 16
--
5
Step – 2 :Reverse of division is multiplication :
X * 5 = 16 * 5 ( multiplying both sides by 5)
--
5
X = 80
Practice :Solve
P + 6 = 11
--
5
X + 24 = 35
--
6
Y + 13 = 42
--
7
Example : Solve for x
x - 14 = 5
--
3
Step -1: Reverse of subtraction is addition.
x - 14 + 14 = 5 +14 ( adding 14 to both sides)
--
3
X = 19
--
3
Step – 2 :Reverse of division is multiplication
X * 3 = 19 * 3 ( multiplying both sides by 3)
---
3
X = 57
Practice :Solve
h - 9 = 18
--
4
X
-- - 4 = 9
7
p
-- - 21 = 5
6
Solving Equations with variables on both sides
In some equations we can see the variables on both sides. First get the variables to one side of the equation and a number on the other side of the equation.
*** Performing the same operation on both sides of an equation does not change the validity of the equation ***
Example : solve 6(b +5) = 48
Use distributive property: a ( b + c ) = a*b + a*c
6(b +5) = 48
6b + 30 = 48 (distributive property)
6b + 30 – 30 = 48 – 30 ( subtracting 30 from both sides)
6b = 18
6b / 6 = 18 / 6 (dividing both sides by 6)
b = 3
Practice:Solve
2( x + 2) = 10
3( x + 4) = 30
5(a – 2) = 0
Example : solve 4a + 3( a – 2) =1
4 a + 3 a – 6 = 1 (distributive property 3 (a – 2) = 3a – 6 )
7a – 6 = 1 ( combining like terms, 4a + 3a = 7a)
7a – 6 + 6 = 1 + 6 ( adding 6 to bothsides)
7a = 7
7a / 7 = 7 / 7 (dividing both sides by 7 )
a = 1
practice: solve
a + 2( a – 7) =1
3x + 4 ( x +2 ) = 22
7p – 3(p + 4 ) = 8
Example : solve 5x = 7 ( x -2)
5x = 7x – 14 (distributive property 7 (x – 2) = 7x – 14 )
5x – 7x = 7x - 7x - 14 (subtracting 7x from both sides to bring x terms one side)
-2x = -14
-2x / -2 = -14 /-2 ( dividing both sides by -2)
X = 7
Practice : solve
3x = 5 ( x -6)
8x = 6( x + 3)
9y = 5( y – 8)
Example: solve 2 ( x + 5 ) = 5 ( x-1)
2 x + 10 = 5x – 5 (distributive property )
2x + 10 – 10 = 5x – 5 – 10 ( subtracting 10 from both sides)
2x = 5x - 15
2x – 5x = 5x – 5x - 15 (subtracting 5x from both sides to bring x terms one side)
- 3 x = - 15
-3x / -3 = - 15 / - 3 ( dividing both sides by -3)
X = 5
Practice : solve
2 ( x - 5 ) = 6 (1- x )
5 ( y + 8 ) = 3 ( y + 20)
6 ( p – 8) = 2 ( 4- p )
*** Performing the same operation on both sides of an equation does not change the validity of the equation ***
Example : solve 6(b +5) = 48
Use distributive property: a ( b + c ) = a*b + a*c
6(b +5) = 48
6b + 30 = 48 (distributive property)
6b + 30 – 30 = 48 – 30 ( subtracting 30 from both sides)
6b = 18
6b / 6 = 18 / 6 (dividing both sides by 6)
b = 3
Practice:Solve
2( x + 2) = 10
3( x + 4) = 30
5(a – 2) = 0
Example : solve 4a + 3( a – 2) =1
4 a + 3 a – 6 = 1 (distributive property 3 (a – 2) = 3a – 6 )
7a – 6 = 1 ( combining like terms, 4a + 3a = 7a)
7a – 6 + 6 = 1 + 6 ( adding 6 to bothsides)
7a = 7
7a / 7 = 7 / 7 (dividing both sides by 7 )
a = 1
practice: solve
a + 2( a – 7) =1
3x + 4 ( x +2 ) = 22
7p – 3(p + 4 ) = 8
Example : solve 5x = 7 ( x -2)
5x = 7x – 14 (distributive property 7 (x – 2) = 7x – 14 )
5x – 7x = 7x - 7x - 14 (subtracting 7x from both sides to bring x terms one side)
-2x = -14
-2x / -2 = -14 /-2 ( dividing both sides by -2)
X = 7
Practice : solve
3x = 5 ( x -6)
8x = 6( x + 3)
9y = 5( y – 8)
Example: solve 2 ( x + 5 ) = 5 ( x-1)
2 x + 10 = 5x – 5 (distributive property )
2x + 10 – 10 = 5x – 5 – 10 ( subtracting 10 from both sides)
2x = 5x - 15
2x – 5x = 5x – 5x - 15 (subtracting 5x from both sides to bring x terms one side)
- 3 x = - 15
-3x / -3 = - 15 / - 3 ( dividing both sides by -3)
X = 5
Practice : solve
2 ( x - 5 ) = 6 (1- x )
5 ( y + 8 ) = 3 ( y + 20)
6 ( p – 8) = 2 ( 4- p )
Tuesday, August 12, 2008
Word problems ( Algebra simple equations)
For solving word problems,first write the expression/equation for the problem.
Example : A number plus 24 is fifty-four. What is the number?
Let the number be x,
X + 24 = 54 is the equation.
X + 24 – 24 = 54 – 24 ( subtracting 24 from both sides)
X = 30
The number is 30.
Practice :
A number plus 35 is seventy four. What is the number?
A number minus 12 is fifty six. What is the number?
A number plus 18 is thirty two. What is the number?
A number minus 25 is sixty eight. What is the number?
Example : Two-sixths of a number equals 44. What is the number?
Let the number be x
2 * x = 44 is the equation
-
6
6/2 * 2/6 * x = 6/2 * 44 (multiplying both sides by 6/2, left side 6/2 and 2/6 gets cancelled)
X = 132 ,The number is 132
Practice :
One -fifths of a number equals 13. What is the number?
Two-thirds of a number equals 62. What is the number?
Three forth of a number equals 36. What is the number?
Example : Twenty-three more than 4 times a number is 71. What is the number?
Let the number be x
23 + 4 x = 71 is the equation.
23 – 23 + 4x = 71 – 23 ( subtracting 23 from both sides)
4x = 48
4x / 4 = 48 / 4
X = 12 ,The number is 12
Practice :
Ten times a number, increased by eighty-three, equals two hundred twenty-three. What is the number?
One hundred fifty-three less than a number is seventy-seven. What is the number?
Three times a number, decreased by 23, equals 34. What is the number?
one-third of a number, decreased by 1 is 8. What is the number?
Two-thirds of a number equals 464. What is the number?
If a number is increased by 6, the result is 76. What is the number?
Negative six times a number is negative one hundred ninety-eight.
What is the number?
Example : A number plus 24 is fifty-four. What is the number?
Let the number be x,
X + 24 = 54 is the equation.
X + 24 – 24 = 54 – 24 ( subtracting 24 from both sides)
X = 30
The number is 30.
Practice :
A number plus 35 is seventy four. What is the number?
A number minus 12 is fifty six. What is the number?
A number plus 18 is thirty two. What is the number?
A number minus 25 is sixty eight. What is the number?
Example : Two-sixths of a number equals 44. What is the number?
Let the number be x
2 * x = 44 is the equation
-
6
6/2 * 2/6 * x = 6/2 * 44 (multiplying both sides by 6/2, left side 6/2 and 2/6 gets cancelled)
X = 132 ,The number is 132
Practice :
One -fifths of a number equals 13. What is the number?
Two-thirds of a number equals 62. What is the number?
Three forth of a number equals 36. What is the number?
Example : Twenty-three more than 4 times a number is 71. What is the number?
Let the number be x
23 + 4 x = 71 is the equation.
23 – 23 + 4x = 71 – 23 ( subtracting 23 from both sides)
4x = 48
4x / 4 = 48 / 4
X = 12 ,The number is 12
Practice :
Ten times a number, increased by eighty-three, equals two hundred twenty-three. What is the number?
One hundred fifty-three less than a number is seventy-seven. What is the number?
Three times a number, decreased by 23, equals 34. What is the number?
one-third of a number, decreased by 1 is 8. What is the number?
Two-thirds of a number equals 464. What is the number?
If a number is increased by 6, the result is 76. What is the number?
Negative six times a number is negative one hundred ninety-eight.
What is the number?
Solving simple equations
An expression is a "phrase" that represents a number.
An equation sets two expressions equal to each other.
In the expression 8x + 17,x is the variable.
8x + 17 = 12 is an equation.
The goal in solving an equation is to get the variable by itself on one side of the equation and a number on the other side of the equation. Performing the same operation on both sides of an equation does not change the validity of the equation
Reversing Operations:(Opposite operations)
Example :slove g + 94 = 176
Here the operation is addition ,reverse of addition is subtraction
g + 94-94 = 176 – 94 ( subtracting 94 from both sides)
g = 18
practice :slove
u + 20 = 90
17 + d = 24
x + 25 = 56
Answers:
u = 70
d = 7
x = 31
Example :slove 2 = n - 8
Here the operation is subtraction, reverse of subtraction is addition
2+8 = n - 8 + 8 ( adding 8 to both sides)
10 = n
practice :slove
x – 89 = 12
y – 21 = 39
p - 33 = 45
Answers:
x = 101
y = 60
p = 78
Example : solve 5x = 30
Here the operation is multiplication(5x means 5 times x), reverse of multiplication is division
5x/5 = 30/5 ( dividing both sides by 5 )
X = 6
practice :solve
8p = 72
4y = 32
9x = 81
Answers:
p = 9
y = 8
x = 9
Example :slove 12 = m ÷ 18
Here the operation is division, reverse of division is multiplication
12 * 18 = (m ÷ 18) * 18 ( multiplying both sides by 18, right side 18 & 18 gets cancelled)
216 = m
practice :slove
a ÷ 4 = 8
P ÷ 7 = 4
x ÷ 8 = 5
Answers:
a = 32
p = 28
x = 40
An equation sets two expressions equal to each other.
In the expression 8x + 17,x is the variable.
8x + 17 = 12 is an equation.
The goal in solving an equation is to get the variable by itself on one side of the equation and a number on the other side of the equation. Performing the same operation on both sides of an equation does not change the validity of the equation
Reversing Operations:(Opposite operations)
Example :slove g + 94 = 176
Here the operation is addition ,reverse of addition is subtraction
g + 94-94 = 176 – 94 ( subtracting 94 from both sides)
g = 18
practice :slove
u + 20 = 90
17 + d = 24
x + 25 = 56
Answers:
u = 70
d = 7
x = 31
Example :slove 2 = n - 8
Here the operation is subtraction, reverse of subtraction is addition
2+8 = n - 8 + 8 ( adding 8 to both sides)
10 = n
practice :slove
x – 89 = 12
y – 21 = 39
p - 33 = 45
Answers:
x = 101
y = 60
p = 78
Example : solve 5x = 30
Here the operation is multiplication(5x means 5 times x), reverse of multiplication is division
5x/5 = 30/5 ( dividing both sides by 5 )
X = 6
practice :solve
8p = 72
4y = 32
9x = 81
Answers:
p = 9
y = 8
x = 9
Example :slove 12 = m ÷ 18
Here the operation is division, reverse of division is multiplication
12 * 18 = (m ÷ 18) * 18 ( multiplying both sides by 18, right side 18 & 18 gets cancelled)
216 = m
practice :slove
a ÷ 4 = 8
P ÷ 7 = 4
x ÷ 8 = 5
Answers:
a = 32
p = 28
x = 40
Writing Expressions for compound statements
Let us discuss about writing Expressions for compound statements
Example : 1 less than the difference of 9 and a number k
Take second part ( bold portion) of the statement, write the expression for it.
1 less than ( k - 9) , now write the expression for this .
(k – 9 ) – 1 = k – 9 – 1 is the expression
practice: Express each phrase as an algebraic expression
14 times the sum of a number y and 6
17 more than the quotient of 6 and a number n
18 plus a number z increased by 8
8 less than 25 multiplied by a number q
7 more than the quotient of 26 and a number d
5 times the sum of a number u and 15
3 times the quotient of 39 and a number n
4 times the sum of a number t and 46
Answers:
14 * (y + 6)
17 + ( 6 / n )
18 + ( z + 8 )
(25 * q ) - 8
7 + ( 26 / d )
5 * ( u + 15)
3 * (39 / n )
4 * ( t + 46 )
Problem : Alex present age is “y” years.
a . What will be his age 5 years from now
b. What was his age 3 years back
c. Alex’s Mother’s age is 4 times his age
d. His Father is 5 years older than Mother
Answer :
y + 5
y - 3
4 * y
5 + (4 * y)
Problem : Let price of sugar per kg be $p,
Write the expression
A . Price of wheat per kg is $ 5 less than price of sugar per kg
B . Price of Ghee per kg is $ 5 times price of sugar per kg
Answers
A. p - 5
B. 5 * p
Example : 1 less than the difference of 9 and a number k
Take second part ( bold portion) of the statement, write the expression for it.
1 less than ( k - 9) , now write the expression for this .
(k – 9 ) – 1 = k – 9 – 1 is the expression
practice: Express each phrase as an algebraic expression
14 times the sum of a number y and 6
17 more than the quotient of 6 and a number n
18 plus a number z increased by 8
8 less than 25 multiplied by a number q
7 more than the quotient of 26 and a number d
5 times the sum of a number u and 15
3 times the quotient of 39 and a number n
4 times the sum of a number t and 46
Answers:
14 * (y + 6)
17 + ( 6 / n )
18 + ( z + 8 )
(25 * q ) - 8
7 + ( 26 / d )
5 * ( u + 15)
3 * (39 / n )
4 * ( t + 46 )
Problem : Alex present age is “y” years.
a . What will be his age 5 years from now
b. What was his age 3 years back
c. Alex’s Mother’s age is 4 times his age
d. His Father is 5 years older than Mother
Answer :
y + 5
y - 3
4 * y
5 + (4 * y)
Problem : Let price of sugar per kg be $p,
Write the expression
A . Price of wheat per kg is $ 5 less than price of sugar per kg
B . Price of Ghee per kg is $ 5 times price of sugar per kg
Answers
A. p - 5
B. 5 * p
Monday, August 11, 2008
Writing Expressions for statements
Writing Expressions for given statements is very important for solving word problems in Algebra.
Writing Expressions for single statements
Example : 2 more than a number p
2 + p is the expression
Example : 5 less than a number x
x - 5 is the expression
Example : 9 times a number a
9 * a is the expression
Example : 23 divided by a number e
23 / e is the expression
Example : difference of a number y and 2
y - 2 is the expression
Example : product of 8 and a number p
8 * p is the expression
Example : One third of a number x
(1/3 )* x = x / 3 is the expression
Practice : Writing Expressions for given statements
44 more than a number n
37 less a number t
product of 24 and a number w
take away a number d from 36
sum of 38 and a number u
quotient of 40 and a number y
a number g decreased by 28
a number f multiplied by 23
45 divided by a number e
difference of a number x and 12
Answers :
44 + n
t - 37
24 * w
36 - d
38 + u
40 / y
g - 28
f * 23
45 / e
x - 12
Writing Expressions for single statements
Example : 2 more than a number p
2 + p is the expression
Example : 5 less than a number x
x - 5 is the expression
Example : 9 times a number a
9 * a is the expression
Example : 23 divided by a number e
23 / e is the expression
Example : difference of a number y and 2
y - 2 is the expression
Example : product of 8 and a number p
8 * p is the expression
Example : One third of a number x
(1/3 )* x = x / 3 is the expression
Practice : Writing Expressions for given statements
44 more than a number n
37 less a number t
product of 24 and a number w
take away a number d from 36
sum of 38 and a number u
quotient of 40 and a number y
a number g decreased by 28
a number f multiplied by 23
45 divided by a number e
difference of a number x and 12
Answers :
44 + n
t - 37
24 * w
36 - d
38 + u
40 / y
g - 28
f * 23
45 / e
x - 12
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