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Wize High School Grade 9 Math Textbook > Solving Equations

Multi-Step Equations with 1 Unknown

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Solving Multi-Step Equations

Sometimes, we need to do more than one step to solve for a variable.


How to solve multi-step equations?

  1. Simplify the left side and right side of the equal sign separately
  2. Expand and simplify any brackets
  3. Group like terms
  4. Deal with   ...   +3\boxed{~~...~~}~\bcth{+3} and   ...   −3\boxed{~~...~~}~\bcth{-3} first
  5. Deal with ×3   ...  \bcfi{\times3}~\boxed{~~...~~} and   ...  3\bcfi{\dfrac{\boxed{~~...~~}}{3}} next
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Example: Solving Multi-Step Equations

ax+b=c        &        xa+b=c\large{\colorOne{ax+b=c~~~~~~~~\&~~~~~~~~\dfrac{x}{a}+b=c}}
Solve for the unknown.

a) 3x+7=133x+7=13

3x+7=13−7−73x3=63x=2\begin{array}{rcr} 3x+7&=&13\\ \scriptsize\colorTwo{-7}&&\scriptsize\colorTwo{-7}\\[1em] \dfrac{3x}{\scriptsize\colorTwo{3}}&=&\dfrac{6}{\scriptsize\colorTwo{3}}\\[1em] x&=&2 \end{array}


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b) 2x−7=52x-7=5

2x−7=5+7+72x2=122x=6\begin{array}{rcr} 2x-7&=&5\\ \scriptsize\colorTwo{+7}&&\scriptsize\colorTwo{+7}\\[1em] \dfrac{2x}{\scriptsize\colorTwo{2}}&=&\dfrac{12}{\scriptsize\colorTwo{2}}\\[1em] x&=&6 \end{array}


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c) 8−3x=148-3x=14

8−3x=14−8             −8−3x−3=6−3x=−2\begin{array}{rcr} 8-3x&=&14\\ \scriptsize\colorTwo{-8}~~~~~~~~~~~~~&&\scriptsize\colorTwo{-8}\\[1em] \dfrac{-3x}{\scriptsize\colorTwo{-3}}&=&\dfrac{6}{\scriptsize\colorTwo{-3}}\\[1em] x&=&-2 \end{array}


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d) y2−4=−6\dfrac{y}{2}-4=-6

y2−4=−6+4+4y2  ×2=−2×2y=−4\begin{array}{rcr} \dfrac{y}{2}-4&=&-6\\ \scriptsize\colorTwo{+4}&&\scriptsize\colorTwo{+4}\\[0.5em] \dfrac{y}{2}~~\scriptsize\colorTwo{\times2}&=&-2\scriptsize\colorTwo{\times2}\\[1em] y&=&-4 \end{array}


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e) 7−y3=57-\dfrac{y}{3}=5

7−y3=5−7            −7−y3  ×−3=−2×−3y=6\begin{array}{rcr} 7-\dfrac{y}{3}&=&5\\ \scriptsize\colorTwo{-7~~~~~~~~~~~~}&&\scriptsize\colorTwo{-7}\\[1em] -\dfrac{y}{3}\scriptsize\colorTwo{~~\times-3}&=&-2\scriptsize\colorTwo{\times-3}\\[1em] y&=&6 \end{array}

Practice: Solving Multi-Step Equations

Solve for the unknown.

a) −2x+5=17-2x+5=17

b) −4a−3=25-4a-3=25

c) 7−2b=−117-2b=-11

d) y3−10=−2\dfrac{y}{3}-10=-2

e) −11−z5=4-11-\dfrac{z}{5}=4
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Example: Solving Multi-Step Equations

ax=b\large\colorOne{ {\dfrac{a}{x}=b}}
a) 12p=−3\dfrac{12}{p}=-3

12p  ×p=−3  ×p12−3=−3p−3−4=pp=−4\begin{array}{rcl} \\\dfrac{12}{p}~~\scriptsize{\colorTwo{\times p}}&=&-3~~\scriptsize{\colorTwo{\times p}}\\[1em] \dfrac{12}{\scriptsize{\colorTwo{-3}}}&=&\dfrac{-3p}{\scriptsize{\colorTwo{-3}}}\\[1em] -4&=&p\\[1em] p&=&-4 \end{array}


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b) −8p=6-\dfrac{8}{p}=6
−8p  ×p=6  ×p−86=6p6−43=pp=−43\begin{array}{rcl} \\\dfrac{-8}{p}~~\scriptsize{\colorTwo{\times p}}&=&6~~\scriptsize{\colorTwo{\times p}}\\[1em] \dfrac{-8}{\scriptsize{\colorTwo{6}}}&=&\dfrac{6p}{\scriptsize{\colorTwo{6}}}\\[1em] -\dfrac{4}{3}&=&p\\[1em] p&=&-\dfrac{4}{3} \end{array}

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c) 5−1p=35-\dfrac{1}{p}=3

5−1p=3−5            −5−1p  ×p=−2  ×p−1−2=−2p−212=pp=12\begin{array}{rcl} 5-\dfrac{1}{p}&=&3\\ \scriptsize{\colorTwo{-5~~~~~~~~~~~~}}&&\scriptsize{\colorTwo{-5}}\\[1em] -\dfrac{1}{p}~~\scriptsize{\colorTwo{\times p}}&=&-2~~\scriptsize{\colorTwo{\times p}}\\[1em] \dfrac{-1}{\scriptsize{\colorTwo{-2}}}&=&\dfrac{-2p}{\scriptsize{\colorTwo{-2}}}\\[1em] \dfrac{1}{2}&=&p\\[1em] p&=&\dfrac{1}{2} \end{array}

Practice: Solving Multi-Step Equations

Isolate the unknown.

a) −20t=−5-\dfrac{20}{t}=-5

b) 21n+5=2\dfrac{21}{n}+5=2
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Example: Solving Multi-Step Equations

ax=b+cx       &       ax+b=cx+d\large{\colorOne{ax=b+cx~~~~~~~\&~~~~~~~ax+b=cx+d}}
Solve the following equations.

a) 3x+8=x3x+8=x

3x+8=x−8     −83x=x−8−x−x2x2=−82x=−4\begin{array}{rcl} 3x+8&=&x\\ \scriptsize\colorTwo{-8}&&\scriptsize\colorTwo{~~~~~-8}\\[1em] 3x&=&x-8\\ \scriptsize\colorTwo{-x}&&\scriptsize\colorTwo{-x}\\[1em] \dfrac{2x}{\scriptsize\colorTwo{2}}&=&\dfrac{-8}{\scriptsize\colorTwo{2}}\\[1em] x&=&-4 \end{array}

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b) −4x+9=−x−15-4x+9=-x-15

−4x+9=−x−15−9              −9−4x=−x−24+x+x−3x−3=−24−3x=8\begin{array}{rcl} -4x+9&=&-x-15\\ \scriptsize\colorTwo{-9}&&\scriptsize\colorTwo{~~~~~~~~~~~~~~-9}\\[1em] -4x&=&-x-24\\ \scriptsize\colorTwo{+x}&&\scriptsize\colorTwo{+x}\\[1em] \dfrac{-3x}{\scriptsize\colorTwo{-3}}&=&\dfrac{-24}{\scriptsize\colorTwo{-3}}\\[1em] x&=&8 \end{array}

Practice: Solving Multi-Step Equations

Solve the following equations.

a) 5x−27=−4x5x-27=-4x

b) −6x+18=5x−15-6x+18=5x-15
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Example: Solving Multi-Step Equations

a(x+b)=c        &        a(bx+c)=d(ex+f)\large\colorOne{a(x+b)=c~~~~~~~~\&~~~~~~~~a(bx+c)=d(ex+f)}
Solve these equations.
a) 4(x−3)=−324(x-3)=-32

To begin, we see that we can simplify a little on the left-hand-side by multiplying through our set of parentheses. At that point each side of the equation will be as simple as we can make it, and we'll already be down to a single instance of our variable xx so we can finish solving by isolating that xx:

4(x−3)=−324x−12=−324x=−32+124x4=−204x=−5\begin{array}{rcl} 4(x-3)&=&-32\\[0.5em] 4x-12&=&-32\\[0.5em] 4x&=&-32\colorTwo{+12}\\[0.5em] \dfrac{4x}{\colorTwo4}&=&\dfrac{-20}{\colorTwo4}\\[1em] x&=&-5 \end{array}

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b) −6(2x+3)=5(3x−9)-6(2x+3)=5(3x-9)

−6(2x+3)=5(3x−9)−12x−18=15x−45−12x=15x−45+18−12x=15x−27−12x−15x=−27−27x=−27−27x−27=−27−27x=1\begin{array}{rcl} -6(2x+3)&=&5(3x-9)\\[0.5em] -12x-18&=&15x-45\\[0.5em] -12x&=&15x-45\colorTwo{+18}\\[0.5em] -12x&=&15x-27\\[0.5em] -12x\colorTwo{-15x}&=&-27\\[0.5em] -27x&=&-27\\[0.5em] \dfrac{-27x}{\colorTwo{-27}}&=&\dfrac{-27}{\colorTwo{-27}}\\[0.5em] x&=&1 \end{array}

Practice: Solving Multi-Step Equations

Solve for the unknown.

a) −3(1+2x)=15-3(1+2x)=15

b) 3(7−2x)=−(20x−7)3(7-2x)=-(20x-7)