Wize High School Grade 9 Math Textbook > Solving Equations

Solving Equations
When we have an equation with 1 variable, we often want to "solve" the equation (a.k.a. "solve for the variable" or "isolate the variable") → We want to rewrite the equation so the variable is by itself on one side of the equal sign.

Golden Rules for Solving Equations
Use the "opposite operation" (inverse) to move something to the other side of the equation
The opposite of addition   ...   +3\boxed{~~...~~}~\bcth{+3}  ...  ​ +3  is subtraction    ...    −3\boxed{~~...~~}~~\bcth{-3}  ...  ​  −3
The opposite of multiplication  ×3    ...  \bm{\colorFive{\times 3}}~~\boxed{~~...~~}×3    ...  ​  is division    ...  3\bcfi{\dfrac{\boxed{~~...~~}}{3}}3  ...  ​​
The opposite of squaring (power of 2)   ...    2\boxed{~~...~~}^{~~\large{\colorTwo2}}  ...  ​  2 is square rooting   ...  \colorTwo {\sqrt {\boxed{~~...~~}}}  ...  ​​
Whatever you do to one side, you MUST do to the other!

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Example: Solving One-Step Equations
Solve the following equations for the unknown variable.

a) x+3=10x+3=10x+3=10

x  +3 =10 −3−3x=7\begin{array}{rcr}
x~~\cancel{+3}~&=&10\\
\scriptsize{~\colorTwo{\cancel{-3}}}&&\scriptsize{\colorTwo{-3}}\\[0.5em]
x&=&7
\end{array}x  +3​  −3​x​==​10−37​

b) y+5=−6y+5=-6y+5=−6

y  +5=−6 −5−5y=−11\begin{array}{rcr}
y~~\cancel{+5}&=&-6\\
\scriptsize{~\colorTwo{\cancel{-5}}}&&\scriptsize{\colorTwo{-5}}\\[0.5em]
y&=&-11
\end{array}y  +5​ −5​y​==​−6−5−11​

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c) x−7=0x-7=0x−7=0

x  −7=0 +7+7x=7\begin{array}{rcr}
x~~\cancel{-7}&=&0\\
\scriptsize{~\colorTwo{\cancel{+7}}}&&\scriptsize{\colorTwo{+7}}\\[0.5em]
x&=&7
\end{array}x  −7​ +7​x​==​0+77​

d) t+(−1)=−6t+(-1)=-6t+(−1)=−6

t  +(−1)=−6 −(−1)−(−1)t=−5\begin{array}{rcr}
t~~\cancel{+(-1)}&=&-6\\
\scriptsize{~\colorTwo{\cancel{-(-1)}}}&&\scriptsize{\colorTwo{-(-1)}}\\[0.5em]
t&=&-5
\end{array}t  +(−1)​ −(−1)​t​==​−6−(−1)−5​

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e) 3x=153x=153x=15

3x3=153x=5\begin{array}{rcr}
\dfrac{\cancel3x}{\scriptsize\colorTwo{\cancel3}}&=&\dfrac{15}{\scriptsize\colorTwo3}\\[1em]
x&=&5
\end{array}3​3​x​x​==​315​5​

f) −4a=16-4a=16−4a=16

−4a−4=16−4a=−4\begin{array}{rcr}
\dfrac{\cancel{-4}a}{\scriptsize\colorTwo{\cancel{-4}}}&=&\dfrac{16}{\scriptsize\colorTwo{-4}}\\[1em]
a&=&-4
\end{array}−4​−4​a​a​==​−416​−4​

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g) x2=−5\dfrac{x}{2}=-52x​=−5

x2   ×2=−5   ×2x=−10\begin{array}{rcr}
\dfrac{x}{\cancel2}~~~\scriptsize{\colorTwo{\cancel{\times2}}}&=&-5~~~\scriptsize{\colorTwo{\times2}}\\[1em]
x&=&-10
\end{array}2​x​   ×2​x​==​−5   ×2−10​

h) −x3=6-\dfrac{x}{3}=6−3x​=6

−x3   ×−3=6   ×−3x=−18\begin{array}{rcr}
\cancel-\dfrac{x}{\cancel3}~~~\scriptsize{\colorTwo{\cancel{\times -3}}}&=&6~~~\scriptsize{\colorTwo{\times-3}}\\[1em]
x&=&-18
\end{array}−​3​x​   ×−3​x​==​6   ×−3−18​

i) −110x=−7-\dfrac{1}{10}x=-7−101​x=−7

−110x   ×−10=−7   ×−10x=70\begin{array}{rcr}
\cancel{-\dfrac{1}{{10}}}x~~~\scriptsize{\colorTwo{\cancel{\times-10}}}&=&-7~~~\scriptsize{\colorTwo{\times-10}}\\[1em]
x&=&70
\end{array}−101​​x   ×−10​x​==​−7   ×−1070​

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j) x=3\sqrt{x}=3x​=3

x=3(x)2=32x=9\begin{array}{rcl}
\sqrt{x}&=&3\\
\left(\sqrt{x}\right)^{\colorTwo 2}&=&3^{\colorTwo 2}\\
x&=&9
\end{array}x​(x​)2x​===​3329​

Practice: Solving One-Step Equations
Solve the following equations.

a) x−10=−4x-10=-4x−10=−4

b) −5z=30-5z=30−5z=30

c) n+9=1n+9=1n+9=1

d) b5=−7\dfrac{b}{5}=-75b​=−7

e) r=10\sqrt{r}=10r​=10

x=

z=

n=

b=

r=

I don't know