Use with Ready Instruction Lesson 15
Dear Family,
Your child is learning about systems of
equations.
A system of two equations is a set of two related equations. The solution of
a system of linear equations can be represented on a graph by the point or
points that both lines have in common. A system of linear equations can
have one solution, no solution, or infinitely many solutions.
y 5 4x 1 1
y 5 4x 1 1
y 5 4.5x 1 1
y 5 3x 1 4
y 5 4x 1 4
y 5 ​  1 ​(9x 1 2)
y
y
2
··
y
14
14
14
12
12
12
10
10
y 5 3x 1 4
8
6
4
2
21 O
22
2
1
2
3
4
8
6
y 5 4x 1 1
4
10
y 5 4x 1 4
8
5
x
21 O
22
24
The lines intersect
and have one point in
common. This point
represents the
solution.
y 5 4.5x 1 1
6
4
y 5 4x 1 1
1
2
3
4
2
5
x
24
The lines are parallel
and have no points in
common. The system
has no solution.
21 O
22
1
2
3
4
5
x
24
The lines are the same
line. There are infinite
solutions because
every point on the line
is a solution to both
equations.
Consider the following example:
How many solutions does this system of linear equations have?
y 5 2x 1 5 and y 5 5x 1 2
On the next page you will see how your child might find the number of
solutions that a system of linear equations has.
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Lesson 15 Understand Systems of Equations
155
Understand Systems of Equations: Sample Solution
How many solutions does this system of linear equations have?
y 5 2x 1 5 and y 5 5x 1 2
One way: Use a graph.
Make a table of values by substituting values for x into each equation
and finding the value of y. Plot the points.
y 5 2x 1 5
x
0
1
2
y
5
7
9
y
30
25
y 5 5x 1 2
20
y 5 5x 1 2
15
x
0
1
2
10
y
2
7
12
5
The graph shows that the lines
intersect at the point (1, 7). The
system has exactly one solution.
21 O
25
y 5 2x 1 5
1
2
3
4
5
x
Another way: Use the equations.
The equations are in the form y 5 mx 1 b, where m is the slope and b is
the y-intercept. Compare the slopes and y-intercepts to determine the
number of solutions the system has.
y 5 2x 1 5
y 5 5x 1 2
slopey-intercept
The equations have different slopes so the system has exactly one
solution regardless of whether the y-intercepts are different or
the same.
Answer: Both methods show that the system has exactly one solution.
156
Lesson 15 Understand Systems of Equations
©Curriculum Associates, LLC Copying is not permitted.