CCSS Math: 8.EE.C.7, 8.EE.C.7b
To solve an equation we find the value of the variable that makes the equation true. For more complicated, fancier equations, this process can take several steps.
When solving an equation, our goal is to find the value of the variable that makes the equation true.

Example 1: Two-step equation

Solve for xx.
3x+7=133x+7=13
We need to manipulate the equation to get xx by itself.
3x+7=133x+77=1373x=63x3=63x=2\begin{aligned} 3x+7&=13 \\\\ 3x+7\redD{-7}&=13\redD{-7} \\\\ 3x&=6 \\\\ \dfrac{3x}{\redD{3}}&=\dfrac{6}{\redD{3}} \\\\ x&=2 \end{aligned}
We call this a two-step equation because it took two steps to solve. The first step was to subtract 77 from both sides, and the second step was to divide both sides by 33. Want an explanation of why we do the same thing to both sides of the equation? Check out this video.
We check the solution by plugging 2\redD2 back into the original equation:
3x+7=1332+7=?136+7=?1313=13       Yes!\begin{aligned} 3x+7&=13 \\\\ 3\cdot \redD 2 + 7 &\stackrel?= 13 \\\\ 6+7 &\stackrel?= 13 \\\\ 13 &= 13 ~~~~~~~\text{Yes!} \end{aligned}

Example 2: Variables on both sides

Solve for aa.
5+14a=9a55 + 14a = 9a - 5
We need to manipulate the equation to get aa by itself.
5+14a=9a55+14a9a=9a59a5+5a=55+5a5=555a=105a5=105a=2\begin{aligned} 5 + 14a &= 9a - 5 \\\\ 5 + 14a \blueD{- 9a} &= 9a - 5 \blueD{- 9a} \\\\ 5 + 5a &= -5 \\\\ 5 + 5a \blueD{-5} &= -5 \blueD{- 5}\\\\ 5a &= -10\\\\ \dfrac{5a}{\blueD5} &= \dfrac{-10}{\blueD5} \\\\ a &= \blueD{-2} \end{aligned}
The answer:
a=2a = \blueD{-2}
Check our work:
5+14a=9a55+14(2)=?9(2)55+(28)=?18523=23       Yes!\begin{aligned} 5 + 14a &= 9a - 5 \\\\ 5 + 14(\blueD{-2}) &\stackrel?= 9(\blueD{-2}) - 5 \\\\ 5 + (-28) &\stackrel?= -18 - 5 \\\\ -23 &= -23 ~~~~~~~\text{Yes!} \end{aligned}
Want to learn more about solving equations with variables on both sides? Check out this video.

Example 3: Distributive property

Solve for ee.
7(2e1)11=6+6e7(2e-1)-11=6+6e
We need to manipulate the equation to get e e by itself.
7(2e1)11=6+6e14e711=6+6e14e18=6+6e14e186e=6+6e6e8e18=68e18+18=6+188e=248e8=248e=3\begin{aligned} 7(2e-1)-11 &= 6+6e \\\\ 14e-7 -11&= 6+6e\\\\ 14e-18 &= 6+6e\\\\ 14e-18\purpleD{-6e} &= 6+6e\purpleD{-6e} \\\\ 8e-18&=6\\\\ 8e-18\purpleD{+18} &=6 \purpleD{+18} \\\\ 8e &=24\\\\ \dfrac{8e}{\purpleD{8}}&= \dfrac{24}{\purpleD{8}}\\\\ e &= \purpleD{3} \end{aligned}
The answer:
e=3 e= \purpleD{ 3 }
Check our work:
7(2e1)11=6+6e7(2(3)1)11=?6+6(3)7(61)11=?6+187(5)11=?243511=?2424=24       Yes!\begin{aligned} 7(2e-1)-11 &= 6+6e \\\\ 7(2(\purpleD{3})-1) -11&\stackrel?= 6+6(\purpleD{3}) \\\\ 7(6-1)-11 &\stackrel?= 6+18 \\\\ 7(5)-11&\stackrel?=24 \\\\ 35-11&\stackrel?=24 \\\\ 24 &=24 ~~~~~~~\text{Yes!} \end{aligned}
Want to learn more about solving equations with the distributive property? Check out this video.

Practice

Problem 1
Solve for bb.
4b+5=1+5b4b+5=1+5b
b=b=
  • 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 these exercises:
Loading