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 $x$.
$3x+7=13$
We need to manipulate the equation to get $x$ by itself.
$\begin{align}
3x+7&=13 \\
3x+7\redD{-7}&=13\redD{-7} \\
3x&=6 \\
\dfrac{3x}{\redD{3}}&=\dfrac{6}{\redD{3}} \\
x&=2
\end{align}$
We call this a two-step equation because it took two steps to solve. The first step was to subtract $7$ from both sides, and the second step was to divide both sides by $3$. 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 $\redD2$ back into the original equation:
$\begin{align}
3x+7&=13 \\
3\cdot \redD 2 + 7 &\stackrel?= 13 \\
6+7 &\stackrel?= 13 \\
13 &= 13 ~~~~~~~\text{Yes!}
\end{align}$
###Example 2: Variables on both sides
Solve for $a$.
$5 + 14a = 9a - 5$
We need to manipulate the equation to get $a$ by itself.
$\begin{align} 5 + 14a &= 9a - 5 \\
5 + 14a \blueD{- 9a} &= 9a - 5 \blueD{- 9a} \\
5 + 5a &= -5 \\
5 + 5a \blueD{-5} &= -5 \blueD{- 5}\\
5a &= -10\\
\dfrac{5a}{\blueD5} &= \dfrac{-10}{\blueD5} \\
a &= \blueD{-2}
\end{align}$
The answer:
$a = \blueD{-2}$
Check our work:
$\begin{align} 5 + 14a &= 9a - 5 \\
5 + 14(\blueD{-2}) &\stackrel?= 9(\blueD{-2}) - 5 \\
5 + (-28) &\stackrel?= -18 - 5 \\
-23 &= -23 ~~~~~~~\text{Yes!}
\end{align}$
Want to learn more about solving equations with variables on both sides? Check out this video.
###Example 3: Distributive property
Solve for $e$.
$7(2e-1)-11=6+6e$
We need to manipulate the equation to get $ e $ by itself.
$\begin{align} 7(2e-1)-11 &= 6+6e \\
14e-7 -11&= 6+6e\\
14e-18 &= 6+6e\\
14e-18\purpleD{-6e} &= 6+6e\purpleD{-6e} \\
8e-18&=6\\
8e-18\purpleD{+18} &=6 \purpleD{+18} \\
8e &=24\\
\dfrac{8e}{\purpleD{8}}&= \dfrac{24}{\purpleD{8}}\\
e &= \purpleD{3}
\end{align}$
The answer:
$ e= \purpleD{ 3 }$
Check our work:
$\begin{align} 7(2e-1)-11 &= 6+6e \\
7(2(\purpleD{3})-1) -11&\stackrel?= 6+6(\purpleD{3}) \\
7(6-1)-11 &\stackrel?= 6+18 \\
7(5)-11&\stackrel?=24 \\
35-11&\stackrel?=24 \\
24 &=24 ~~~~~~~\text{Yes!}
\end{align}$
Want to learn more about solving equations with the distributive property? Check out this video.
Practice
[[☃ graded-group-set 1]]
Want more practice? Check out these exercises: