Intro. To Graphing Linear Equations
The Coordinate Plane
A. The coordinate plane has 4 quadrants.
B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate
(the ordinate). The point is stated as an ordered pair (x,y).
C. Horizontal Axis is the X – Axis. (y = 0)
D. Vertical Axis is the Y- Axis (x = 0)
Plot the following points:
a) (3,7)
b) (-4,5)
c) (-6,-1)
d) (6,-7)
e) (5,0)
f) (0,5)
g) (-5,0)
f) (0, -5)
y-axis
x-axis
1
Slope Intercept Form
Before graphing linear equations, we need to be familiar with slope intercept form. To understand slope
intercept form, we need to understand two major terms: The slope and the y-intercept.
Slope (m):
The slope measures the steepness of a non-vertical line. It is sometimes referred to as the rise over run. It’s
how fast and in what direction y changes compared to x.
y-intercept:
The y-intercept is where a line passes through the y axis. It is always stated as an ordered pair (x,y). The x
coordinate is always zero. The y coordinate can be found by plugging in 0 for the X in the equation or by
finding exactly where the line crosses the y-axis.
What are the coordinates of the y-intercept line pictured in the diagram above? :
Some of you have worked with slope intercept form of a linear equation before. You may remember:
y = mx + b
Using y = mx + b, can you figure out the equation of the line pictured above?:
2
Graphing Linear Equations
Graphing The Linear Equation:
y = 3x - 5
1) Find the slope:
m=3m= 3 . = y .
1
x
2) Find the y-intercept: x = 0 , b = -5  (0, -5)
3) Plot the y-intercept
4) Use slope to find the next point: Start at (0,-5)
m = 3 . = ▲y .  up 3 on the y-axis
1
▲x
 right 1 on the x-axis
(1,-2) Repeat: (2,1) (3,4) (4,7)
5) To plot to the left side of the y-axis, go to y-int. and
do the opposite. (Down 3 on the y, left 1 on the x)
(-1,-8)
6) Connect the dots.
Do Now:
1) y = 2x + 1
2) y = -4x + 5
3
3) y = ½ x – 3
4) y= - ⅔x + 2
5) y = -x – 3
6) y= 5x
4
Q3 Quiz 1 Review
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
1) y = 4x - 6
2) y = -2x + 7
5
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
3) y = -x - 5
4) y = 5x + 5
6
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
5) y = - ½ x - 7
6) y = ⅗x - 4
7
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
7) y = ⅔x
8) y = - ⅓x + 4
8
Finding the equation of a line in slope intercept form
(y=mx + b)
Example: Using slope intercept form [y = mx + b]
Find the equation in slope intercept form of the line formed by (1,2) and (-2, -7).
A. Find the slope (m):
m = y2 – y1
x2 – x1
B. Use m and one point to find b:
y = mx + b
m= 3
x= 1
y= 2
m = (-7) – (2) .
(-2) – (1)
2 = 3(1) + b
2=3+b
-3 -3
-1 = b
m = -9 .
-3
y = 3x – 1
m= 3
Example: Using point slope form [ y – y1 = m(x – x1) ]
Find the equation in slope intercept form of the line formed by (1,2) and (-2, -7).
A. Find the slope (m):
m = y2 – y1
x2 – x1
B. Use m and one point to find b:
y – y1 = m(x – x1)
m= 3
x= 1
y= 2
y – (2) = 3(x – (1))
m = (-7) – (2) .
(-2) – (1)
m = -9 .
-3
m= 3
y – 2 = 3x - 3
+2
+2
y = 3x – 1
9
Find the equation in slope intercept form of the line formed by the given points. When you’re finished, graph
the equation on the give graph.
1) (4,-6) and (-8, 3)
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2) (4,-3) and (9,-3)
III.
Special Slopes
A. Zero Slope
* No change in Y
* Equation will be Y =
* Horizontal Line
3) (7,-2) and (7, 4)
B. No Slope (undefined slope)
* No change in X
* Equation will be X =
* Vertical Line
11
Point-Slope Form
 y – y1 = m(x – x1)
Slope Intercept Form  y = mx + b
Standard Form:
 Ax + By = C
 “y” is by itself
 Constant (number) is by itself
Given the slope and 1 point, write the equation of the line in: (a) point-slope form,
(b) slope intercept form, and (c) standard form:
Example: m = ½ ; (-6,-1)
a) Point-Slope Form
b) Slope intercept form
c) Standard Form
1) m = -2; (-3,1)
a) Point-Slope Form
b) Slope intercept form
c) Standard Form
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2) m = - ¾ ; (-8, 5)
Point-Slope Form
b) Slope intercept form
c) Standard Form
3) m = ⅔; (-6, -4)
Point-Slope Form
b) Slope intercept form
c) Standard Form
4) m = -1 (5, -1)
Point-Slope Form
b) Slope intercept form
c) Standard Form
13
Find equation in slope intercept form and graph:
1) (3,-2)(-6,-8)
3) (3,7) (3,-7)
2) (-6,10) (9,-10)
4) (7,-6)(-3,4)
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5) (5,-9)(-5,-9)
6) m= 4 (-2,-5)
7) m= ⅔ (-6,-7)
8) m= -
(8,-1)
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9) m = 0 (4,3)
10) m = undefined (-6, 5)
11) 16x -4y =36
12) 8x+24y = 96
16
13) y+7=2(x+1)
14) y+5=(2/5)(x+10)
15) y-7= ¾ (x-12)
16) y-2=-3(x-2)
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Q3 Quiz 2 Review
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
1) y - 2 = -3(x – 1)
2) 14x + 21y = -84
18
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
3) y + 10 = 5(x + 2)
4) y – 7 = ¼ (x – 20)
19
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
5) 8x – 8y = 56
6) y + 6 = -1(x – 4)
20
Find the equation in slope intercept form and graph:
If your printer doesn’t print the graphs, you must use
your own graph paper.
7) 18x – 12y = -12
8) y – 15 = (-5/3)(x + 9)
Answers: 1) y = -3x + 5
5) y = x - 7
2) y = - ⅔ x - 4
6) y = - x – 3
3) y = 5x
7) y = (3/2)x - 1
4) y = ¼ x + 2
8) y = -(5/3)x
21
Graph both of the lines on the same set of axis:
y = -2x + 6
y = -2x – 5
IV. Parallel and Perpendicular Lines:
A. Parallel Lines
* Do not intersect
* Have same slopes
For the given line, find a line that is parallel and passes through the given point and graph
Given Line:
Parallel:
Given Line:
Parallel:
7) y = ⅓ x + 4
(6,1)
8) y = 4x – 5
(2,13)
Given Line:
9) y = -⅔ x + 2
Parallel:
(-9,2)
Given Line:
10) y = –5x + 6
Parallel:
(4,-27)
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Practice Problems: a) Use the two points to find the equation of the line.
b) For the line found in part a, find a line that is parallel and passes through the given
point.
c) Graph both lines on the same set of axis.
Given Line:
1) (-5, 13) (3, -3)
Parallel:
(4,-10)
Given Line:
2) (-6,0) (3,6)
Parallel:
(6,3)
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Given Line:
3) (2,6)(-3,-19)
Parallel:
(5,30)
Given Line:
4) (-4,3) (-8,6)
Parallel:
(-4, 10)
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Given Line:
5) (2,-5) (-2, -5)
Parallel:
(8,-2)
Given Line:
6) (-9,-11)(6,9)
Parallel:
(-3,-9)
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Given Line:
7) (8,-3) (-4,9)
Parallel:
(-2, 1)
Given Line:
8) (3,6)(3,-6)
Parallel:
(7,-3)
26
Given Line:
9) (4,-3)(-6,-8)
Parallel:
(6,7)
Given Line:
10) (2,4)(-6,-12)
Parallel:
(-3,-5)
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11) Find the equation of the line parallel to y = 3x – 2, passing through (-2, 1).
12) Find the equation of the line parallel to y = -½x – 5, passing through (-2, 7)
13) Find the equation of the line parallel to y = -¼ x + 2, passing through (-8, 4)
14) Find the equation of the line parallel to y = (3/2)x + 6, passing through (-6, -11)
15) Find the equation of the line parallel to y = -5, passing through (2,7)
16) Find the equation of the line parallel to x = 5, passing through (6, -4).
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Q3 Quiz 3 Review
FOLLOW REQUIRED FORMAT AND SHOW ALL PROPER WORK!
a) Use the two points to find the equation of the line.
b) For the line found in part a, find a line that is parallel and passes through the given point.
c) Graph both lines on the same set of axis.
Given Line:
1) (-4, 13) (3, -8)
Parallel:
(4,-17)
Given Line:
2) (8,1) (-4,-5)
Parallel:
(-6,2)
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Given Line:
3) (5,4) (-4,4)
Parallel:
(-6,-7)
For #’s 4-7, just find the equation. You do not have to graph.
4) Find the equation of the line parallel to y = -⅗x – 2, passing through (-5, 7).
5) Find the equation of the line parallel to y = 4x – 5, passing through (-4, 9)
6) Find the equation of the line parallel to y = 2, passing through (-8, -9)
7) Find the equation of the line parallel to x = 5, passing through (-6, -11)
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Solving Systems of Equations
Graphically
A system of equations is a collection of two or more equations with a same set of unknowns. In solving a
system of equations, we try to find values for each of the unknowns that will satisfy every equation in the
system. When solving a system containing two linear equations there will be one ordered pair (x,y) that will
work in both equations.
To solve such a system graphically, we will graph both lines on the same set of axis and look for the point of
intersection. The point of intersection will be the one ordered pair that works in both equations. We must then
CHECK the solution by substituting the x and y coordinates in BOTH ORIGINAL EQUATIONS.
1) Solve the following system graphically:
y = 2x – 5
y = - ⅓x + 2
31
Solve each of the systems of equations graphically:
2) y + 1 = -3(x – 1)
7x + 7y = 42
3) y – 9 = ¾ (x – 12)
6x + 12y = -60
32
4) 12x – 8y = 48
y – 4 = -2(x – 2)
5) y + 5 = 2(x + 4)
y – 10 = - ½ (x + 4)
33
Solve each system graphically and check:
6)
y = -4x -5
y = 2x -7
7)
6x + 3y =21
12x + 16y = -48
34
8)
12x – 6y = -6
16x -8y = 40
9)
y= -4
x=7
35
10)
y-2= (3/5)(x-10)
y+11 =2(x+7)
11)
6x + 9y = 45
9x +15y = 75
36
12)
x=5
y-12 = -3(x+2)
13)
9x – 18y = 126
y = -4
37
Q3 Quiz 4 Review
1) y = 2x + 6
y = - ½x - 4
2) 15x - 15y = -45
y - 2 = -3(x + 1)
38
3) y + 3 = (-2/3)(x - 3)
y + 1 = -1(x + 2)
4) 24x - 18y = -18
y = -7
39
5) y + 2 = (-3/4)(x - 8)
17x - 34y = 204
6) x = -6
y + 15 = (5/3)(x + 9)
40
7) y - 5 = ¼(x - 4)
45x - 15y = 105
8) 24x - 12y = -72
y – 2 = 2(x + 2)
41
9) 11x + 44y = -176
y - 3 = - ¼ (x + 4)
10) y + 4 = (2/3)(x + 12)
25x + 50y = -150
Answer Key:
1) (-4,-2)
2) (-1,2)
6) (-6,-10)
7) (4,5)
3) (-6,3)
8) Many Solutions
4) (-6, -7)
9) No Solution
5) (8,-2)
10) (-6,0)
42
Graphing Inequalities
When we solved and graphed inequalities with only one variable (ex: x > 3), we moved on to compound
inequalities (AND/OR). We would graph both inequalities on the same number line and decide what to keep
based on whether it was an AND or an OR problem. When we graphed linear equations on the coordinate plane
we moved on to solving systems of equations graphically.
When we graph inequalities in two variables on the coordinate plane, we do not graph compound inequalities.
We move on to solving systems of inequalities. It takes a little from both inequalities with one variable and
solving systems graphically.
Graph the Inequality:
y>¼x+3
Step 1: Graph the line.
y>¼x+3
m = ¼ = ▲y = up 1
▲x
r4
y-int= (0,3)
Step 2: Test a point one up from the from
the y-int and one down from the y-int):
(0, 2)
2 > ¼ (0)+3
(0, 4)
4 > ¼ (0) + 3
2 > 3
4>3
FALSE
TRUE
Step 3: Shade towards the “true” point (0,4)
When you “test”, you must do
it in the original inequality!
43
1) 6x - 9y > 36
2) y - 3 > -2(x + 1)
44
3) 12x + 9y < 27
4) y + 4 > -3(x - 3)
45
5) y > 4
6) x < -6
46
Q3 Quiz 5 Review
1) 72x – 216y < -432
2) y + 1 > ⅖ (x + 10)
47
3) y – 5 < - ½ (x + 10)
4) 48x + 12y < -48
48
5) x > 7
6) y < -2
49
7) x < -4
8) y > 6
50
Graphing Systems of Inequalities
Solve the system of inequalities graphically:
y>¼x+3
y < 3x – 5
Step 1: Graph the 1st inequality
y>¼x+3
m=¼=
y-int= (0,3)
Step 2: Graph the 2nd inequality
y < 3x – 5
m = 3/1 =
y-int.= (0,-5)
(0, 2)
(0-6)
TEST
(0, 4)
TEST
(0,-4)
2 > ¼ (0)+3
-3 > ¼ (0) + 3
-6 < 3(0) - 5
-4 < 3(0) - 5
2 > 3
4>3
-6 < -5
-4 < -5
FALSE
TRUE
TRUE
FALSE
Step 3: Label the area where the shading intersects with an “S”
51
2) y - 3 < - ⅓(x – 6)
12x – 6y > -12
3) x > 4
y < -5
52
4) 24x + 6x > -6
y>2
5) y – 6 < ⅔(x - 9)
x < -3
53
Q3 Quiz 6 Review
1) y > 5
3x - y > -3
2) y + 6 > -½ (x - 8)
y - 4 > 2(x – 2)
54
3) 15x – 45y < 90
x>3
4) 21x – 7y > 14
y- 3 > -¼ (x + 12)
55