Download Systems of Linear Equations: Solving by Graphing

8.1
Systems of Linear Equations:
Solving by Graphing
8.1
OBJECTIVE
1. Find the solution(s) for a set of linear equations
by graphing
From our work in Section 6.1, we know that an equation of the form x y 3 is a linear
equation. Remember that its graph is a straight line. Often we will want to consider two
equations together. They then form a system of linear equations. An example of such a
system is
xy3
3x y 5
A solution for a linear equation in two variables is any ordered pair that satisfies the equation. Often there is just one ordered pair that satisfies both equations of a system. It is called
the solution for the system. For instance, there is one solution for the system above, and it
is (2, 1) because, replacing x with 2 and y with 1, we have
xy3
3x y 5
213
3215
33
615
55
NOTE There is no other
ordered pair that satisfies both
equations.
Because both statements are true, the ordered pair (2, 1) satisfies both equations.
One approach to finding the solution for a system of linear equations is the graphical
method. To use this, we graph the two lines on the same coordinate system. The coordinates of the point where the lines intersect is the solution for the system.
Example 1
Solving by Graphing
Solve the system by graphing.
xy6
xy4
NOTE Use the intercept
method to graph each
equation.
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NOTE By substituting 5 for x
First, we determine solutions for the equations of our system. For x y 6, two solutions
are (6, 0) and (0, 6). For x y 4, two solutions are (4, 0) and (0, 4). Using these intercepts, we graph the two equations. The lines intersect at the point (5, 1).
y
and 1 for y into the two
original equations, we can
check that (5, 1) is indeed the
solution for our system.
xy6
xy4
5 1 6
514
66
44
Both statements must be true
for (5, 1) to be a solution for
the system.
xy4
(5, 1)
x
(5, 1) is the solution of the system.
It is the only point that lies on both lines.
xy6
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CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
CHECK YOURSELF 1
Solve the system by graphing.
y
2x y 4
xy5
x
Example 2 shows how to graph a system when one of the equations represents a
horizontal line.
Example 2
Solving by Graphing
Solve the system by graphing.
3x 2y 6
y6
For 3x 2y 6, two solutions are (2, 0) and (0, 3). These represent the x and y intercepts
of the graph of the equation. The equation y 6 represents a horizontal line that crosses the
y axis at the point (0, 6). Using these intercepts, we graph the two equations. The lines will
intersect at the point (2, 6). So this is the solution to our system.
y
y6
x
3x 2y 6
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622
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY GRAPHING
SECTION 8.1
623
CHECK YOURSELF 2
Solve the system by graphing.
y
4x 5y 20
y 8
x
The systems in Examples 1 and 2 both had exactly one solution. A system with one
solution is called a consistent system. It is possible that a system of equations will have
no solution. Such a system is called an inconsistent system. We present such a system
here.
Example 3
Solving an Inconsistent System
Solve by graphing.
2x y 2
2x y 4
We can graph the two lines as before. For 2x y 2, two solutions are (0, 2) and (1, 0).
For 2x y 4, two solutions are (0, 4) and (2, 0). Using these intercepts, we graph the two
equations.
y
NOTE In slope-intercept form,
2x y 2
our equations are
y 2x 2
and
y 2x 4
Both lines have slope 2.
x
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2x y 4
Notice that the slope for each of these lines is 2, but they have different y intercepts. This
means that the lines are parallel (they will never intersect). Because the lines have no points
in common, there is no ordered pair that will satisfy both equations. The system has no
solution. It is inconsistent.
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CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
CHECK YOURSELF 3
Solve by graphing.
x 3y 3
x 3y 6
y
x
There is one more possibility for linear systems, as Example 4 illustrates.
Example 4
Solving a Dependent System
Solve by graphing.
x 2y 4
NOTE Notice that multiplying
2x 4y 8
the first equation by 2 results in
the second equation.
Graphing as before and using the intercept method, we find
y
The two equations have
the same graph!
x 2y 4
2x 4y 8
Because the graphs coincide, there are infinitely many solutions for this system. Every
point on the graph of x 2y 4 is also on the graph of 2x 4y 8, so any ordered pair
satisfying x 2y 4 also satisfies 2x 4y 8. This is called a dependent system, and
any point on the line is a solution.
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x
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY GRAPHING
SECTION 8.1
CHECK YOURSELF 4
Solve by graphing.
y
x y4
2x 2y 8
x
The following summarizes our work in this section.
Step by Step:
Step 1
Step 2
To Solve a System of Equations by Graphing
Graph both equations on the same coordinate system.
Determine the solution to the system as follows.
a. If the lines intersect at one point, the solution is the ordered pair
corresponding to that point. This is called a consistent system.
y
x
A consistent system
NOTE There is no ordered pair
that lies on both lines.
b. If the lines are parallel, there are no solutions. This is called an
inconsistent system.
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y
x
An inconsistent system
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CHAPTER 8
SYSTEMS OF LINEAR EQUATIONS
c. If the two equations have the same graph, then the system has
infinitely many solutions. This is called a dependent system.
NOTE Any ordered pair that
corresponds to a point on the
line is a solution.
y
x
A dependent system
Step 3
Check the solution in both equations, if necessary.
CHECK YOURSELF ANSWERS
1.
xy5
2.
y
y
(5, 8)
y8
(3, 2)
x
x
4x 5y 20
2x y 4
3. There is no solution. The lines are parallel, so the system is inconsistent.
y
x
x 3y 3
x 3y 6
4.
x y4
2x 2y 8
y
x
A dependent system
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626
Name
8.1
Exercises
Section
Date
Solve each of the following systems by graphing.
1. x y 6
2. x y 8
xy4
ANSWERS
xy2
1.
y
y
2.
3.
4.
x
x
5.
6.
3. x y 3
4.
xy5
xy7
x y 3
y
y
x
x
5. x 2y 4
6. 3x y 6
x y1
xy4
y
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y
x
x
627
ANSWERS
7.
7. 2x y 8
8. x 2y 2
2x y 0
8.
x 2y 6
y
9.
y
10.
11.
x
x
12.
x 3y 12
2x 3y 6
10. 2x y 4
2x y 6
y
y
x
x
11. 3x 2y 12
12.
y3
y
y
x
628
x 2y 8
3x 2y 12
x
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9.
ANSWERS
13.
x y4
2x 2y 8
14. 2x y 8
x2
y
13.
y
14.
15.
16.
x
x
17.
18.
15. x 4y 4
16. x 6y 6
x 2y 8
x y 4
y
y
x
x
17. 3x 2y 6
18. 4x 3y 12
2x y 5
x y2
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y
y
x
x
629
ANSWERS
19.
19. 3x y 3
20. 3x 6y 9
3x y 6
20.
x 2y 3
y
y
21.
22.
23.
x
x
24.
21.
2y 3
x 2y 3
22.
x y 6
x 2y 6
y
y
x
23. x 4
x
24. x 3
y 6
y5
y
y
x
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x
630
ANSWERS
25. Find values for m and b in the following system so that the solution to the system is
25.
(1, 2).
26.
mx 3y 8
3x 4y b
27.
28.
26. Find values for m and b in the following system so that the solution to the system is
29.
(3, 4).
5x 7y b
mx y 22
30.
27. Complete the following statements in your own words:
“To solve an equation means to . . . .”
“To solve a system of equations means to . . . .”
28. A system of equations such as the one below is sometimes called a “2-by-2” system
of linear equations.”
3x 4y 1
x 2y 6
Explain this term.
29. Complete this statement in your own words: “All the points on the graph of the
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equation 2x 3y 6 . . . .” Exchange statements with other students. Do you agree
with other students’ statements?
30. Does a system of linear equations always have a solution? How can you tell without
graphing that a system of two equations graphs into two parallel lines? Give some
examples to explain your reasoning.
631
ANSWERS
a.
Getting Ready for Section 8.2 [Section 1.6]
b.
Simplify each of the following expressions.
c.
(a)
(c)
(e)
(g)
d.
e.
(2x y) (x y)
(3x 2y) (3x 3y)
2(x y) (3x 2y)
3(2x y) 2(3x y)
(b)
(d)
(f )
(h)
(x y) (x y)
(x 5y) (2x 5y)
2(2x y) (4x 3y)
3(2x 4y) 4(x 3y)
f.
g.
Answers
1. x y 6
xy4
3. x y 3
(5, 1)
xy5
(1, 4)
y
y
x
x
5. x 2y 4
x y1
7. 2x y 8
(2, 1)
2x y 0
y
(2, 4)
y
x
632
x
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h.
9.
x 3y 12
2x 3y 6
11. 3x 2y 12
(6, 2)
y3
y
(2, 3)
y
x
13.
x
x y4
Dependent
2x 2y 8
15. x 4y 4
x 2y 8
(4, 2)
y
y
x
x
17. 3x 2y 6
2x y 5
19. 3x y 3
(4, 3)
3x y 6
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y
Inconsistent
y
x
x
633
21.
2y 3
x 2y 3
0,
3
2
23. x 4
y 6
(4, 6)
y
y
x
25. m 2, b 5
27.
c. y
e. 5x
a. 3x
29.
f. 5y
g. 5y
b. 2y
h. 10x
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d. 3x
x
634