Equivalent systems of equationsÂ review

CCSS Math: HSA.REI.C.5
Two systems of equations are equivalent if they have the same solution(s). This article reviews how to tell if two systems are equivalent.
Systems of equations that have the same solution are called equivalent systems.
Given a system of two equations, we can produce an equivalent system by replacing one equation by the sum of the two equations, or by replacing an equation by a multiple of itself.
In contrast, we can be sure that two systems of equations are not equivalent if we know that a solution of the one is not a solution of the other.
Note: This idea of equivalent systems of equations pops up again in linear algebra. However, the examples and explanations in this article are geared to a high school algebra 1 class.

Example 1

We're given two systems of equations and asked if they're equivalent.
System ASystem B
\begin{aligned}-12x+9y=7\\\\9x-12y=6\end{aligned}\begin{aligned}-12x+9y=7\\\\3x-4y=2\end{aligned}
If we multiply the second equation in System B by $3$, we get:
\begin{aligned} 3x-4y&=2 \\\\ 3(3x-4y)&=3(2) \\\\ 9x-12y&=6 \end{aligned}
Replacing the second equation of System B with this new equation, we get an equivalent system:
\begin{aligned}-12x+9y=7\\\\9x-12y=6\end{aligned}
Whoa! Look at that! This system is the same as System A, which means system A is equivalent to System B.

Example 2

We're given two systems of equations and asked if they're equivalent.
System ASystem B
\begin{aligned}-9x-4y&=5\\\\2x+5y&=-4\end{aligned}\begin{aligned}-7x+y&=1\\\\2x+5y&=-4\end{aligned}
Interestingly, if we sum the equations in System A, we get:
Replacing the first equation in System A with this new equation, we get a system that's equivalent to System A:
\begin{aligned}-7x+y&=1\\\\2x+5y&=-4\end{aligned}
Lo and behold! This is System B, which means that System A is equivalent to System B.

Example 3

We're given two systems and asked to prove that they aren't equivalent by finding a solution of one that is not a solution of the other.
System ASystem B
\begin{aligned}-4x+10y&=1\\\\-1x-2y&=-3\end{aligned}\begin{aligned}-9x-y&=8\\\\-1x-2y&=4\end{aligned}
Notice how the coefficients for $x$ and $y$ in the second equations of both systems are the same. However, the constant terms in the two equations are different!
Whichever pair of values for $x$ and $y$ that makes System A true will make System B false, and vice versa.
For example, $x=1$, $y=1$ is a solution to the second equation in System A, but it's not a solution to the second equation in System B.
System A and System B are not equivalent.

Practice

Problem 1
Elsa and Olaf's teacher gave them a system of linear equations to solve. They each took a few steps that led to the systems shown in the table below.
Teacher
$5x+3y=-1$
$4x-9y=8$
ElsaOlaf
$4x-9y=8$$15x+9y=-3$
$9x-6y=7$$4x-9y=-5$
Which of them obtained a system that is equivalent to the teacher's system?
Remember that two linear systems are "equivalent" if they have the same solution.