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
Wednesday, August 20, 2008
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
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