Solve for “y” for each:
1.) 2x + y = 9
2.) -x = y + 3
-2x
-2x
y = -2x + 9
-3
y = -x - 3
3.) 2y – 4 = 3x
+4
+4
2y = 3x + 4
-3
4.) 2 = y – ⅓x
3
y = x+2
2
5.) 7x – y = -8
-7x
-7x
-y = -7x - 8 y = 7x + 8
+⅓x
+⅓x
y = ⅓x + 2
6.) 3y – 1 = -3x
+1
+11
y = -x +
3y = -3x + 1
3
Again, solve for “y”
1.) 2x + 2y = 10
- 2x
Divide by 2
2.) 6x – 3y = -3
- 2x
2y = -2x + 10
y = -x + 5
3.) 2y – 7 = -x
+7
+7
2y = -1x + 7 Divide by 2
Divide by -3
- 6x
- 6x
-3y = -6x - 3
-3
-3 -3
y = 2x + 1
4.) 15x + 5y = 20
- 15x
- 15x
5y = -15x + 20
Divide by 5
y = -½x + 3.5
y = -3x + 4
3.3 Graphing
Linear
Equations
Linear Equation
An equation for which the
graph is a STRAIGHT
line
NO FRACTIONS and NO LEADING
NEGATIVES!!!
Linear Equation: an equation
whose graph forms a line.
In linear equations, all variables are
taken to the first power.
3x  4 y  11
y  x  6x  2
2
is linear.
Why?
is not.
What is a Linear Equation?
y
A linear equation is an equation
whose graph is a LINE.
x
Linear
Not Linear
Solution
Any ordered pair of numbers that makes a
linear equation true.
FOR: y = x - 9
(x,y)
(9,0) IS ONE SOLUTION
Can you think of another solution?
Is (11,2)??
Is (-5,-14)??
Yes
Yes
INFINITE
AMOUNT OF
SOLUTIONS
Determine if the ordered
pair given is a solution of
x + 2y = 5
1.) (1,2)
2.) (7,-3)
(1) + 2(2) = 5??
(7) + 2(-3) = 5??
5=5
YES
1=5
NO
Steps to GRAPHING
1.) SOLVE FOR “y”
2.) MAKE A T-CHART
3.) GRAPH EACH ORDERED PAIR
4.) CONNECT THE DOTS TO DRAW
THE LINE
Linear Equation
Example:
y=x+3
Graphing
Find solutions using table
y=x+3
x
y
x
y
-2
-2
1
-1
-1
2
0
0
3
1
1
4
2
2
5
3
3
6
Create these ordered pairs
(-2,1) and (-1,2) and (0,3)
and (1,4) and (2,5) and (3,6)
Graphing
Graph the points from the table
(-2,1) (-1,2) (0,3) (1,4) (2,5)
Graphing
Draw a line to connect them
So by looking at this
line we can assume
that (5,8) is a
solution?
What other points can we
conclude is a solution???
y = x +3
Suppose you were asked to graph
the solution set of the linear
equation, x + y = 3
FIRST SOLVE FOR Y!!
y = -x + 3
x+y=3
-x
-x
y = -x + 3
Then creating a table of values.
x
y
Use the x-values -2, -1, 0, 1, and 2
y = -x+3
x
y
-2
5
-1
4
0
1
2
3
2
1
With your table completed for three
sets of values, you are ready to
graph the line.
y = -x+3
x
y
-2
5
-1
4
y = -x + 3
0
1
2
3
2
1
Graphing a Linear Equation
y
How do we graph linear equations?
Let’s try this one: y = 3x – 2
Make a Table of values
x
x
y
–2
–8
y = 3(–2) – 2 = –8
–1
–5
y = 3(–1) – 2 = –5
0
–2
y = 3(0) – 2 = –2
1
1
y = 3(1) – 2 = 1
2
4
y = 3(2) – 2 = 4
Graphing a Linear Equation
y
How about another one!
Let’s try x – 2y = 6.
First Step:
x – 2y = 6
Solve for y
–2y = -x + 6
y = ½x - 3
Second Step: Make a t-chart
x
y
–4
-5
½(-4) – 3 = -5
–2
-4
½(-2) – 3 = -4
0
-3
2
-2
4
-1
½(-0) – 3 = -3
x
Create a table of values for each
equation, and graph it on grid paper:
x
y
1. x  y  5 -2 3 2. 2 x  y  7
y=x+5
x
y
-2
11
6
-1
9
2
7
0
7
3
8
1
5
2
3
3
1
-1
4
0
5
1
3. y  2 x  4
y =-2 x + 7
x
y
x
y
-2
-8
-4
-1
-1
-6
-2
0
0
-4
0
1
1
-2
2
2
2
0
4
3
3
2
6
4
1
4. y  x  1
2
Try These
Make a table for each function
using x-valuse -2, -1, 0, 1, 2, 3:
1) y = 2x + 3
2) y = -x - 4
x
y
x
y
-2
-1
-2
-2
-1
1
-1
-3
0
3
0
-4
1
5
1
-5
2
7
2
-6
3
9
3
-7
Find some solutions of this
linear equations
1.) -2x + y = -3
First Solve for y
y = 2x - 3
Use the x-values of
-1, 0, 1, 2 and 3
x
y
-1
-5
0
-3
1
-1
2
1
3
3
Make a t-chart for this
function:
1.) -4x + y = -1
Graph the function y = 3x -2
First make a t-chart
Graph the function: 4y – 2x = 8
First solve for “y”
4y – 2x = 8
+2x +2x
4y = 2x + 8
4 4
4
y = ½x + 2
Then use which
x-values???
x
y
-4
0
-2
1
0
2
2
3
4
4
6
5
Graph the function: 3x – 2y = 6
First solve for “y”
3x – 2y = 6
-3x
-3x
-2y = -3x + 6
-2
-2 -2
y = 3/2 x - 3
Then use which
x-values???
x
y
-4
-9
-2
-6
0
-3
2
0
4
3
6
6
Graph the
equation
y = 2x + 6
X
Y = 2x + 6
Y
(x, y)
1
2
3
4
5
2(1) + 6
2(2) + 6
2(3) + 6
2(4) + 6
2(5) + 6
8
10
12
14
16
(1, 8)
(2, 10)
(3, 12)
(4, 14)
(5, 16)
Graph the equation: -6x + 2y = 4
y = 3x + 2
Plot (-2, -4), (0, 2) and (1, 5)
Then draw the line. Make sure your
line covers the graph and has
arrows on both ends. Be sure to
use a ruler.
Which ordered
pair does NOT
satisfy the
equation
2x + y = 7?
Which ordered is a
solution of the
equation
5x – 3y = 17?
a.) (1,5)
a.) (4,1)
b.) (3,2)
b.) (1,4)
c.) (0,7)
c.) (5,1)
d.) (4,-1)
d.) (1,5)