MA 12B
Arcadia Valley Career Technology Center
Topic: Linear Equations and Systems
Show-Me Standards: MA1,MA4
Tech Math I Credit
Last Update: March 2008
Focus: Slope/Equation Forms of Lines
MO Grade Level Expectations: : N2C8,
N2C9, A2B7
NCTM Standards: 4A, 5A
OBJECTIVE: The students will be able to find the slope of a line and recognize an
equation of a line in Slope Intercept Form, Point-Slope Form, and Standard Form. They
will also be able to graph an equation of a line using the slope and y-intercept.
Terms:
Slope- a ratio that defines the steepness of a line. The steepness is measured by comparing the
change of a line’s vertical movement to the change of its horizontal movement. Also written as:
Rise or change in y coordinates
Run
change in x coordinates
Slope is denoted by the letter “m” . Since the y coordinates of the pairs on the line determine the
line’s vertical movement and the x coordinates determine the line’s horizontal movement, we can
develop a formula that will always give you the slope ratio if you can find, or be given, two points on
the line.
Slope Formula- (x1,y1) and (x2,y2) are two points on the same line.
m = y2 – y1 _
x2 – x1
If y2 – y1 = 0 , then the line has zero slope.
If x2 – x1 = 0 , then the line has no slope.
Zero Slope- all horizontal lines have zero slope. They do not have any vertical movement.
No Slope- (also called undefined slope) all vertical lines have no slope. They do not have any
horizontal movement.
Positive Slope- if the slope ratio is positive, then the line is rising from left to right on a graph.
Negative Slope- if the slope ratio is negative, then the line is falling from left to right on a graph.
EQUATION FORMS OF A LINE:
Slope-Intercept Form of a Line- any equation that can be written in the form y = mx + b , where
m is the slope and b is the value of the y-intercept.
Point-Slope Form of a Line- any equation that can be written in the form y – y1 = m(x – x1) ,
where m is the slope and (x1,y1) , is a point on the line.
Standard Form of a Line- any equation that can be written in the form Ax + By = C , where
A,B, and C are integers(positive or negative whole numbers) and A and B are not both zeros.
Look at this graph and the information following:
y
^
3
-------- ●
3
2|
-------- ●
(6,2)
3
2|
----------- ●
2|
(3,0)
|●
(0,-2)
(9,4)
x
slope = rise = m = 2/3
run
y-intercept = b = -2
because (0,-2) is the point where the line crosses the y-axis.
Slope-Intercept Form of the Line:
y = 2/3 x + -2 or
Point-Slope Form of the Line: y – 4 = 2/3 ( x – 9)
Standard Form of the Line: 2x + -3y = 6
or
y = 2/3 x – 2
note: we could have used any one
of the points on the line for
the (x1,y1)
2x – 3y = 6
-Notice there are several different ways to write the equation that forms the line. The equations all
look different, but they all represent the same line.
-Notice that if you work the Slope Formula with any two points on the line, you get 2/3.
(3,0) and (9,4)
(3,0) and (6,2)
(0,-2) and (6,2)
m = 4 – 0_ = 4/6 = 2/3
9–3
m = 2 – 0_ = 2/3
6–3
m = 2 – -2 = 4/6 = 2/3
6–0
-The larger the absolute value of the slope ratio , the steeper the line regardless whether it rises or
falls. Let’s look at the slope numbers 2/3 , 5/3 , and -7 .
‫ ׀‬2/3 ‫ = ׀‬2/3
‫ ׀‬5/3 ‫ = ׀‬5/3
‫ – ׀‬7 ‫ = ׀‬7 = 7/1
2/3 means, as the line rises two units on the coordinate graph, it runs horizontally three units.
The previous graph shows the steepness of slope 2/3. The line rises from left to right because
the slope is positive even before you do the absolute value.
5/3 means, as the line rises five units on the graph, it runs horizontally three units. This would
make a steeper line than the line with slope 2/3. The five makes the line rise quicker than the
example above. The line rises from left to right because the slope is positive even before you
do the absolute value.
-7 means, this line is the steepest of the three. It has the largest absolute value. 7/1 is greater
than 2/3 and 5/3. This line falls seven units as it runs one unit. It falls because the slope is
negative before the absolute value is applied.
Here is a quick picture of the lines that contain the slopes above:
-you can see that the -7
slope is the steepest.
2/3
5/3
-7
Examples:
Find the slope of the line that contains the two points.
EX1. (7,4) and ( 8,6)
m = 6 – 4_ = _2_ = 2
8–7
1
EX2. (-3,-2) and (4½,-1)
m = -1 – -2 = 1_ =
4½ - -3
7½
2_
15
Identify the slope and y-intercept in the following linear equations.
EX3. y = 6x +3
Recognizing the Slope-Intercept Form,
m=6 ,b=3
EX5. 3y = 2x – 7
Rearranging into Slope-Intercept Form,
3y_ = 2x_ – 7_
3
3
3
y = 2/3 x – 7/3
m = 2/3 , b = -7/3
EX6. y – 3 = ½ ( x – 5)
Recognizing Point-Slope Form,
m=½
Rearranging into Slope-Intercept Form,
y – 3 = ½x – 5/2
y – 3 + 3 = ½x – 5/2 + 3
y = ½x + ½
b=½
EX8. 5x – 6y = 0
Rearranging into Slope-Intercept Form,
5x – 5x – 6y = 0 – 5x
-6y = -5x
- 6y_ = -5x_
-6
-6
y = 5/6 x
m = 5/6 , b = 0
EX4. -8 = 2y + 4x
Rearranging into the SlopeIntercept Form,
-8_ = 2y_ + 4x_
2
2
2
-4 = y + 2x
-4 – 2x = y + 2x – 2x
-4 – 2x = y or y = -2x – 4
m = -2 , b = -4
EX7. 4x + 6y = 18
Recognizing Standard Form,
Rearranging into SlopeIntercept Form,
4x – 4x + 6y = 18 – 4x
6y = 18 – 4x
6y_ = 18_ – 4x_
6
6
6
y = 3 – 2/3x or y = -2/3x + 3
m = -2/3 , b = 3
EX9. y = 8
EX10. x = -5
Recognize a Horizontal Line,
m=0 ,b=8
Recognize a Vertical Line,
m = no slope , b = no y-intercept
Graph the following using the slope and y-intercept of each line.
EX11. 8x – 2y = 12
Recognize Standard Form,
Rearrange into Slope-Intercept Form,
8x – 8x – 2y = 12 – 8x
-2y = 12 – 8x
-2y_ = 12_ – 8x_
-2
-2
-2
y = -6 – - 4x
y = -6 + 4x or y = 4x – 6
m = 4 or 4/1 and b = -6
y
^
● (1,-2)
● (0,-6)
-if you start at the y-intercept of -6. The point is (0,-6).
Go up four, because the slope is positive causing a rise,
and over one. This lands you at the point (1,-2). You
could continue going up four and over one if you wished
to create several more points on the line.
>
x
EX12. y – 5 = -1/2 (x – 3)
Recognize Point-Slope Form,
Rearrange into Slope-Intercept Form,
y – 5 = -1/2 x – -3/2
y – 5 = -1/2 x + 3/2
y – 5 + 5 = -1/2 x + 3/2 + 5
y = -1/2 x + 6 ½
m = -1/2 and b = 6 ½
y
^
●
(0,6 ½)
●
(2,5 ½)
>
x
-if you start at the y-intercept 6 ½. The point is (0,6 ½).
Go down one, because the slope is negative causing a fall,
and over two. This lands you at the point (2,5 ½).
EX13. x = 4
Recognize a Vertical Line,
m = no slope and b = no y-intercept
y ^
● (4,7)
● (4,5)
● (4,3)
● (4,1)
● (4,-1)
● (4,-3)
-no slope and all points have 4 as their x-coordinate.
>
x
Problems:
Tell which equation form the following linear equations are given in. Find the slope and y-intercept
of each.
1. y = 4x + 7
2. y – 3 = ½(x + 6)
3. 7x + 5y = 12
4. 5x + -1 = y
5. 7/2(x – 9) = y – 4
6. 2x – 4y = 8
Find the slope of the line that runs through the given points using the Slope Formula.
7. (9,-7) and (7,-8)
8. (1,3) and (0,-4)
9. ( 5,-6) and (2,-6)
10. (2½,7) and (4½,8)
Tell whether the line is horizontal or vertical and give its slope if it has one.
11. 6 = y
12. x = -10
13. y – 5 = ½(0x – 3)
14. 7x = 28
Graph the following using the slope and y-intercept of each line.
15. y = 3x – 4
16. y – 1 =⅜(x + -8)
17. 2y = 10
18. x = -7
19. y = -½ x + 8
20. 20x + 4y = 8