copyright © 2013 pearson education, inc. section 3.2 linear equations in two variables
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Copyright © 2013 Pearson Education, Inc.
Section 3.2
Linear Equations in Two Variables
Example
Determine whether the given ordered pair is a solution to the given equation.a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)
Solution
a. y = x + 57 = 2 + 57 = 7 True
The ordered pair (2, 7) is a solution.
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Example
Determine whether the given ordered pair is a solution to the given equation.a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)
Solution
b. 2x + 3y = 18
2(3) + 3(4) = 18
6 12 = 18
6 18
The ordered pair (3, 4) is NOT a solution.
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ExampleA table can be used to list solutions to an equation.
Complete the table for the equation y = 3x – 1.
Solution
x 3 1 0 3
y
3x
3 1
3( 3) 1
9 1
10
y x
y
y
y
10
1x
3 1
3( ) 1
3
4
1
1
y x
y
y
y
0x
3 1
3( ) 1
0
1
0
1
y x
y
y
y
3x
3 1
3( ) 1
9
8
3
1
y x
y
y
y
4 1 8
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Find five solutions to the equation y = 3x + 2.
Choose some x values and then compute the corresponding y values and complete the table.
If x = -2, y = 3(-2) + 2 = -4. Ordered pair
If x = -1, y = 3(-1) + 2 = -1. Ordered pair
If x =0, y = 3(0) + 2 = 2. Ordered pair
If x =1, y = 3(1) + 2 = 5. Ordered pair
If x =2, y = 3(2) + 2 = 8. Ordered pair
Graph on next slide.
Finding Solutions of an Equation
X Y (x,y)
-2
-1
0
1
2
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)4,2(4
)1,1(1
)2,0(2
)5,1(5
)8,2(8
A table that lists a few solutions is helpful when graphing an equation
Plot the five ordered pairs to obtain the graph of y = 3x + 2
Graph of the Equation
X Y (X,Y)
-2 -4 (-2,-4)
-1 -1 (-1,-1)
0 2 (0,2)
1 5 (1,5)
2 8 (2,8)
Example
Make a table of values for the equation y = 3x, and then use the table to graph this equation.SolutionStart by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table. x y
–1 –3
0 0
1 3
2 6Plot the points and connect the points with a straight line.
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A linear equation in two variables can be written as
Ax + By = C,
where A, B, and C are fixed numbers (constants) and A and B are not both equal to 0.
The graph of a linear equation in two variables is a line.
Linear Equation in Two VariablesPage 172
Example
Graph the linear equation. SolutionBecause this equation can be written in standard form, it is a linear equation.
Choose any three values for x.
11
4y x
x y
–4 0
0 1
4 2
Plot the points and connect the points with a straight line.
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Example
Graph the linear equation. SolutionBecause this equation can be written in standard form, it is a linear equation.
Choose any three values for x.
Plot the points and connect the points with a straight line.
5x y
x y
0 5
2 3
5 0
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Example
Graph the linear equation by solving for y first.
SolutionSolve for y.
3 6 12x y
3 6 12x y
6 3 12y x 1
22
y x
x y
–2 1
0 2
2 3
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Plot the five ordered pairs to obtain the graph of y = 3x + 1
(2,7)
(1,4)
(0,1)
(-1,-2)
(-2,-5)
Graph of the Equation
X Y (x,y)
2 7 (2,7)
1 4 (1,4)
0 1 (0,1)
-1 -2 (-1,-2)
-2 -5 (-2,-5)
DONE
Tables of Solutions
A table can be used to list solutions to an equation.
A table that lists a few solutions is helpful when graphing an equation.
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Basic Concepts
Equations can have any number of variables.
A solution to an equation with one variable is one number that makes the statement true.
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Page 172
Example
Graph the linear equation by solving for y first.
SolutionSolve for y.
3 6 12x y
3 6 12x y
6 3 12y x 1
22
y x
x y
–2 1
0 2
2 3
Page 174
Plot the ordered pairs to obtain the graph of y = 2x
Graph of the Equation
2
)1(2
y
y
0
)0(2
y
y
4
)2(2
y
y
4
)2(2
y
y
Plot the ordered pairs to obtain the graph of y = 2x-2
Graph of the Equation
0
2)1(2
y
y
2
2)0(2
y
y
2
2)2(2
y
y
6
2)2(2
y
y
Plot the ordered pairs to obtain the graph of
Graph of the Equation
22
1 xy
3
2)2(2
1
y
y
2
2)0(2
1
y
y
4
2)4(2
1
y
y
1
2)2(2
1
y
y
Plot the five ordered pairs to obtain the graph of y = 3x + 2
Graph of the Equation
X Y (x,y)
-2 -4 (-2,-4)
-1 -1 (-1,-1)
0 2 (0,2)
1 5 (1,5)
2 8 (2,8)