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Solve for “y ” for each:
1.) 2x + y = 9 2.) -x = y + 3
3.) 2y – 4 = 3x 4.) 2 = y – ⅓x
5.) 7x – y = -8 6.) 3y – 1 = -3x
y = -2x + 9
y = ⅓x + 2
y = -x - 3
y = 7x + 8 3y = -3x + 1
+⅓x +⅓x
-2x -2x - 3 - 3
+4 +4
-7x -7x +1 +1
-y = -7x - 8
2y = 3x + 4
Again, solve for “y ”
1.) 2x + 2y = 10 2.) 6x – 3y = -3
3.) 2y – 7 = -x 4.) 15x + 5y = 20
y = -x + 5y = 2x + 1
y = -½x + 3.5 y = -3x + 4
- 2x - 2x
2y = -2x + 10Divide by 2
- 6x - 6x
-3y = -6x - 3Divide by -3
+ 7 + 7
2y = -1x + 7 Divide by 2
- 15x - 15x
5y = -15x + 20
Divide by 5
-3 -3 -3
3.3 Graphing Linear Equations
Linear EquationAn equation for which the
graph is a STRAIGHT line
NO FRACTIONS and NO LEADING NEGATIVES!!!
Linear Equation: an equation whose graph forms a line.
1143 yx
262 xxy
is linear.
is not.
In linear equations, all variables are taken to the first power.
Why?
What is a Linear Equation?
A linear equation is an equation whose graph is a LINE.
y
x
Linear
Not Linear
SolutionAny ordered pair of numbers that makes a
linear equation true.
FOR: y = x - 9
(9,0) IS ONE SOLUTION
Can you think of another solution?
Is (11,2)??
Is (-5,-14)??
Yes
Yes
INFINITE AMOUNT OF SOLUTIONS
(x,y)
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??
YES NO
5 = 5 1 = 5
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
GraphingFind solutions using table
y = x + 3
x y
-2
-1
0
1
2
3
x y
-2 1
-1 2
0 3
1 4
2 5
3 6
Create these ordered pairs
(-2,1) and (-1,2) and (0,3)
and (1,4) and (2,5) and (3,6)
GraphingGraph the points from the table
(-2,1) (-1,2) (0,3) (1,4) (2,5)
GraphingDraw 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,
FIRST SOLVE FOR Y!!
Then creating a table of values.xy
x + y = 3-x -x
y = -x + 3
Use the x-values -2, -1, 0, 1, and 2
xy
-2 -1 0 1 2
5 4 3 2 1
With your table completed for three sets of values, you are ready to graph the line.
xy
-2 -1 0 1 2
5 4 3 2 1
y = -x + 3
Graphing a Linear Equation
How do we graph linear equations?
Let’s try this one: y = 3x – 2
Make a Table of values
x y
–2
–1
0
1
2
y = 3(–2) – 2 = –8
y = 3(–1) – 2 = –5
y = 3(0) – 2 = –2
y = 3(1) – 2 = 1
y = 3(2) – 2 = 4
–8
–5
–2
1
4
y
x
Graphing a Linear Equationy
x
How about another one!
Let’s try x – 2y = 6.
First Step:
Solve for y
x – 2y = 6
–2y = -x + 6
y = ½x - 3
Second Step: Make a t-chart
x y
–4
–2
0
2
4
-5
-4
-3
-2
-1
½(-4) – 3 = -5
½(-2) – 3 = -4
½(-0) – 3 = -3
Create a table of values for each equation, and graph it on grid paper:
1. 2.
3.
5 yx 72 yx
42 xy4. 1
21 xy
x y
-2 3
-1 4
0 5
1 6
2 7
3 8
y = x + 5
x y
-4 -1
-2 0
0 1
2 2
4 3
6 4
x y
-2 -8
-1 -6
0 -4
1 -2
2 0
3 2
x y
-2 11
-1 9
0 7
1 5
2 3
3 1
y =-2 x + 7
Try TheseMake a table for each function using x-
valuse -2, -1, 0, 1, 2, 3:1) y = 2x + 3 2) y = -x - 4
x y
-2 -1
-1 1
0 3
1 5
2 7
3 9
x y
-2 -2
-1 -3
0 -4
1 -5
2 -6
3 -7
Find some solutions of this linear equations
1.) -2x + y = -3
First Solve for y
y = 2x - 3x y
-1 -5
0 -3
1 -1
2 1
3 3
Use the x-values of-1, 0, 1, 2 and 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 = 8First solve for “y”
4y – 2x = 8 +2x +2x
4y = 2x + 84 4 4
y = ½x + 2 x y
-4 0
-2 1
0 2
2 3
4 4
6 5
Then use which x-values???
First solve for “y”
3x – 2y = 6-3x -3x
-2y = -3x + 6-2 -2 -2
y = 3/2 x - 3 x y
-4 -9
-2 -6
0 -3
2 0
4 3
6 6
Then use which x-values???
Graph the function: 3x – 2y = 6
Graph the equationy = 2x + 6
X Y = 2x + 6 Y (x, y)
1 2(1) + 6 8 (1, 8) 2 2(2) + 6 10 (2, 10) 3 2(3) + 6 12 (3, 12) 4 2(4) + 6 14 (4, 14) 5 2(5) + 6 16 (5, 16)
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.
Graph the equation: -6x + 2y = 4
y = 3x + 2
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)
b.) (3,2)
c.) (0,7)
d.) (4,-1)
a.) (4,1)
b.) (1,4)
c.) (5,1)
d.) (1,5)