7.1
Solving Linear Systems
by Graphing
Goals p Solve a system of linear equations by graphing.
p Model a real-life problem using a linear system.
VOCABULARY
System of linear equations Two or more linear equations in
the same variables form a system of linear equations, or
simply a linear system.
Solution of a system of linear equations A solution of a system
of linear equations in two variables x and y is an ordered
pair (x, y) that satisfies each equation in the system.
SOLVING A LINEAR SYSTEM USING GRAPH-AND-CHECK
To use the graph-and-check method to solve a system of linear
equations in two variables, use the following steps.
Step 1 Write each equation in a form that is easy to graph .
Step 2 Graph both equations in the same coordinate plane .
Step 3 Estimate the coordinates of the point of intersection .
Step 4 Check the coordinates algebraically by substituting into
each equation of the original linear system.
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Example 1
Using the Graph-and-Check Method
Solve the linear system graphically. Check the solution algebraically.
5x 4y 12
Equation 1
3x 4y 20
Equation 2
Solution
1. Write each equation in a form that is easy to graph, such as
slope-intercept form.
Equation 1
Equation 2
5x 4y 12
3x 4y 20
4y 5x 12
4y 3x 20
5
4
3
4
y x 3
5
4
y x 5
3
4
slope: slope: y-intercept: 3
y-intercept: 5
2. Graph these equations.
3x 4y 20 y
5
3. The two lines appear to intersect
at ( 4 , 2 ).
4. To check ( 4 , 2 ) as a solution
algebraically, substitute 4 for x
and 2 for y in each original
equation.
Equation 1
3
1
5
3
1
1
x
5x 4y 12
3
Equation 2
5x 4y 12
3x 4y 20
5( 4 ) 4( 2 ) 12
3( 4 ) 4( 2 ) 20
20 8 12
12 8 20
12 12
20 20
Answer Because ( 4 , 2 ) is a solution of each equation in the
linear system, ( 4 , 2 ) is a solution of the linear system.
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Checkpoint Graph and check to solve the linear system.
1. 3x 4y 4
2. 5x 2y 4
x 2y 8
9x 2y 12
y
y
7
5
1
3
1
1
3
5
7 x
2
4 x
1
1
1
3
5
7 x
3
(2, 3)
(4, 2)
3. y 2x 3
4. y 3x 4
2x 5y 25
7x 3y 6
y
y
2
7
4
5
2
2
3
4
1
7
5
3
1
1 x
(5, 7)
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6
(3, 5)
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