Substitution method review (systems of equations)

The substitution method is a technique for solving a system of equations. This article reviews the technique with multiple examples and some practice problems for you to try on your own.

What is the substitution method?

The substitution method is a technique for solving systems of linear equations. Let's walk through a couple of examples.

Example 1

We're asked to solve this system of equations:
3x+y=3x=y+3\begin{aligned} 3x+y &= -3\\\\ x&=-y+3 \end{aligned}
The second equation is solved for xx, so we can substitute the expression y+3-y+3 in for xx in the first equation:
3x+y=33(y+3)+y=33y+9+y=32y=12y=6 \begin{aligned} 3\blueD{x}+y &= -3\\\\ 3(\blueD{-y+3})+y&=-3\\\\ -3y+9+y&=-3\\\\ -2y&=-12\\\\ y&=6 \end{aligned}
Plugging this value back into one of our original equations, say x=y+3x = -y +3, we solve for the other variable:
x=y+3x=(6)+3x=3\begin{aligned} x &= -\blueD{y} +3\\\\ x&=-(\blueD{6})+3\\\\ x&=-3 \end{aligned}
The solution to the system of equations is x=3x=-3, y=6y=6.
We can check our work by plugging these numbers back into the original equations. Let's try 3x+y=33x+y = -3.
3x+y=33(3)+6=?39+6=?33=3\begin{aligned} 3x+y &= -3\\\\ 3(-3)+6&\stackrel ?=-3\\\\ -9+6&\stackrel ?=-3\\\\ -3&=-3 \end{aligned}
Yes, our solution checks out.

Example 2

We're asked to solve this system of equations:
7x+10y=362x+y=9\begin{aligned} 7x+10y &= 36\\\\ -2x+y&=9 \end{aligned}
In order to use the substitution method, we'll need to solve for either xx of yy in one of the equations. Let's solve for yy in the second equation:
2x+y=9y=2x+9\begin{aligned} -2x+y&=9 \\\\ y&=2x+9 \end{aligned}
Now we can substitute the expression 2x+92x+9 in for yy in the first equation of our system:
7x+10y=367x+10(2x+9)=367x+20x+90=3627x+90=363x+10=43x=6x=2 \begin{aligned} 7x+10\blueD{y} &= 36\\\\ 7x+10\blueD{(2x+9)}&=36\\\\ 7x+20x+90&=36\\\\ 27x+90&=36\\\\ 3x+10&=4\\\\ 3x&=-6\\\\ x&=-2 \end{aligned}
Plugging this value back into one of our original equations, say y=2x+9y=2x+9, we solve for the other variable:
y=2x+9y=2(2)+9y=4+9y=5\begin{aligned} y&=2\blueD{x}+9\\\\ y&=2\blueD{(-2)}+9\\\\ y&=-4+9 \\\\ y&=5 \end{aligned}
The solution to the system of equations is x=2x=-2, y=5y=5.
Want to learn more about the substitution method? Check out this video.

Practice

Problem 1
Solve the following system of equations.
5x+4y=3x=2y15\begin{aligned} -5x+4y &= 3\\\\ x&=2y-15 \end{aligned}
x= x=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y=y=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Want more practice? Check out this exercise.
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