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Use with Ready Instruction Lesson 16
Dear Family,
Your child is learning to solve systems of
equations algebraically.
A system of equations is a set of related equations. A system
of linear equations can have one solution, no solution, or
infinitely many solutions.
y
8
7
One way to find the solution to a system is to graph each
equation. The point where the lines meet is the solution.
y5x13
6
5
4
The graph shows the solution of the system:
3
y 5 2x
y5x13
1
y 5 2x
2
21 O
21
The solution is the point of intersection, (3, 6).
1
2
3
4
5
x
Another way to solve a system of equations is by manipulating the
equations algebraically, using either substitution or elimination.
These are the steps for solving a system using substitution.
• Rewrite one equation (if necessary) in a form that sets y equal to an
expression involving x.
• Substitute the expression for y into the other equation.
• Solve that equation to find the value of the second variable, x.
• Substitute the value of x into either original equation to solve for y.
• Check your answer.
Consider the following example:
Solve this system of equations.
y5x12
2y 5 2x 1 13
On the next page you will see how your child will use substitution and
another method called elimination to solve this system.
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Lesson 16 Solve Systems of Equations Algebraically
163
Solve Systems of Equations Algebraically:
Sample Solution
Solve this system of equations.
y 5 x 1 2
2y 5 2x 1 13
One way: Use substitution.
The first equation is y 5 x 1 2, so there’s no need to rewrite.
Solve for x.
Substitute x 1 2 for y in the second equation.
2(x 1 2) 5 2x 1 13 2x 1 4 5 2x 1 13 Multiply.
3x 1 4 5 13 Add x to both sides.
3x 5 9 Subtract 4 from both sides.
x 5 3 Divide both sides by 3.
Solve for y.
y 5 (3) 1 2 5 5 Substitute 3 for x in the first equation.
The solution to the system is (3, 5).
Another way: Use elimination.
The coefficients of x are 1 and 21. They are opposites, which means
their sum is zero, so adding the equations will eliminate the x-terms.
y 5 x 1 2
2y 5 2x 1 13
y 1 2y 5 x 2 x 1 2 1 13 Add the equations.
3y 5 15 Simplify.
y 5 5 Divide both sides by 3.
5 5 x 1 2 Substitute 5 for y in the first equation.
3 5 x Solve for x.
The solution to the system is (3, 5).
Answer: Both methods show that the solution to the system is (3, 5).
164
Lesson 16 Solve Systems of Equations Algebraically
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