Solving Two-Step Equations

Solving Two-Step Equations exercise

• Determine the equation describing the situation.

  Hints

  If one of your favorite band's album costs $~\$12$ then the total cost of buying four of albums is $4\times \$12=\$48$.

  The total battery power she needs to take photos and record the video should be equal to the remaining power.

  Solution

  How many photos can Shannon take and still have enough battery power left to record a video?

  • First we assign the unknown number of photos to the variable $x$.
  • One photo uses $2\%$ of the battery so at most, Shannon can take $x\times 2$ photos.
  • Because the video requires $8\%$ of the battery, the total battery life Shannon needs is $x\times 2\%+8\%$ (without the percent signs).
  • On the other side of the equation, we write the remaining battery life: $30\%$.

  Doing so gives us the following equation:

  $x\times 2\%+8\%=30\%$.

  To solve the equation, we leave out the $\%$ sign:

  $x\times 2+8=30$

  We can solve the equation using the following steps:

  $\begin{array}{rcl} x\times 2+8 & = & ~30\\ \color{#669900}{-8} & &\color{#669900}{-8}\\ x\times 2 & = & ~22 \\ \color{#669900}{\div2} & & \color{#669900}{\div2}\\ x & = & ~11 \end{array}$

• Solve the equation.

  Hints

  We have to do two steps:

  1. add or subtract and
  2. multiply or divide.

  Use the opposite operator:

  • $+~\longleftrightarrow~-$ and
  • $\times~\longleftrightarrow~\div$.

  Imagine an equation as a scale in balance.

  Removing a weight from only one side leads to an imbalance.

  Solution

  Look at the two steps for solving equations:

  1. add or subtract and
  2. multiply or divide.

  To solve the equation $x\times 2+8=30$ we first subtract $8$ on both sides of the equation. This is the opposite of addition: $x\times 2=22$.

  Next we divide by $2$ on both sides of the equation. This is the opposite of multiplication:

  $x=11$.

  So, Shannon could take 11 photos and still have enough power left to record a video.

• Determine how many photos Shannon and her friends can take.

  Hints

  The equation looks quite similar to $x\times 2+8=30$.

  Think about what the numbers mean and what you have to change.

  Use opposite operations to solve the equation.

  Think about whether the number of photos you can take increases or decreases if you have more battery power.

  Solution

  The equations we use for each friend's situation look closely resemble $x\times 2+8=30$.

  Anne

  $\begin{array}{rcl} x\times 2+8 &=&~20\\ \color{#669900}{-8} & & \color{#669900}{-8}\\ x\times 2 &=& ~12 \\ \color{#669900}{\div2} & & \color{#669900}{\div2}\\ x &=&~~~6 \end{array}$

  Anne can take six photos.

  George

  $\begin{array}{rcl} x\times 2+8 &=&~24\\ \color{#669900}{-8} & &\color{#669900}{-8}\\ x\times 2 &=&~16 \\ \color{#669900}{\div2} & & \color{#669900}{\div2}\\ x &=&~~~8 \end{array}$

  George can take eight photos.

  Linda

  $\begin{array}{rcl} x\times 2+8 &=&~32\\ \color{#669900}{-8} & & \color{#669900}{-8}\\ x\times 2 &=&~24\\ \color{#669900}{\div 2} & &\color{#669900}{\div 2}\\ x &=&~12 \end{array}$

  Linda can take twelve photos.

  Paul

  $\begin{array}{rcl} x\times 2+8 &=&~40\\ \color{#669900}{-8} & & \color{#669900}{-8}\\ x\times 2 &=&~32\\ \color{#669900}{\div 2} & & \color{#669900}{\div 2}\\ x &=&~16 \end{array}$

  Paul can take sixteen photos.

• Solve the following equations.

  Hints

  Use two steps:

  1. Simplify the equation.
  2. Isolate the variable by using opposite operations.

  Each solution is a natural number.

  You can check your solution by substituting your answer for $x$ back into the equation.

  Solution

  Each of the following equation will be solved in the same manner. We use two steps:

  1. Simplify the equation.
  2. Isolate the variable by using opposite operations.

  Which operation we use depends on the operation in the equation.

  First equation

  $\begin{array}{rcl} x\div 4-2&=&~~~1\\ \color{#669900}{+2}&&\color{#669900}{+2}\\ x\div 4&=&~~~3\\ \color{#669900}{\times 4}&& \color{#669900}{\times 4}\\ x&=&~12 \end{array}$

  Second equation

  $\begin{array}{rcl} x\times 4-2&=&~10\\ \color{#669900}{+2}&&\color{#669900}{+2}\\ x\times 4&=&~12\\ \color{#669900}{\div4}&=&\color{#669900}{\div4}\\ x&=&~~~3 \end{array}$

  Third equation

  $\begin{array}{rcl} x\times 5+3&=&~~123\\ \color{#669900}{-3}&&\color{#669900}{~-3}\\ x\times 5&=&~~120\\ \color{#669900}{\div5}&&\color{#669900}{~\div5}\\ x&=&~~~~24 \end{array}$

  Fourth equation

  $\begin{array}{rcl} x\div 5+3&=&~~~4\\ \color{#669900}{-3}&&\color{#669900}{-3}\\ x\div 5&=&~~~1\\ \color{#669900}{\times 5}&&\color{#669900}{\times 5}\\ x&=&~~~5 \end{array}$

• Describe how to solve two-step equations.

  Hints

  Use Opposite Operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Solution

  To solve algebraic equations, you must isolate the variable by using Opposite Operations.

  For solving two-step equations, you have to use Opposite Operations twice. First use Opposite Operations to remove the constant then again to remove the coefficient to the variable.

  Use Opposite Operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Let's have a look at the following equation:

  $3\times x-4=8$.

  Step 1

  • Since we have $-4$ on the left side of the equation, we should add $4$ on both sides of the equation to get $3\times x=12$.

  Step 2

  • In order to isolate the variable $x$, we use the Opposite Operation of multiplication, which is division. So we divide by $3$ on both sides of the equation to get $x=4$.

• Set up an equation to model the situation.

  Hints

  To write an equation for each store, write the total amount Ben and his siblings can spend, $200$, on the right side of the equal sign. On the left side of the equal sign, write the sum of the cost of the game system, and the unknown number of games times the cost of each game.

  Use two steps:

  1. Simplify the equation.
  2. Isolate the variable by using opposite operations.

  For example if Ben and his siblings went to J&S E-Games, like they always do, they could have bought the game system for $\$112.00$ and each game for $\$16.00$.

  If we set up the equation, we get:

  $x\times \$16.00+\$112.00=\$200$

  1. Subtract $\$112.00$: $x\times \$16.00=\$88.00$
  2. Divide by $\$16.00$: $x=5$

  Solution

  First we need an equation that describes the situation:

  $\text{number of games}\times \text{price per game} + \text{game system price} =200$.

  Remember, we're calling the number of games Ben and his siblings can buy $x$.

  Tami's Toys

  • $\$138.00$ for the game system
  • $\$15.50$ for each game

  $\begin{array}{rcr} x\times 15.50+138.00&=&~~~200.00\\ \color{#669900}{-138.00}&&\color{#669900}{-138.00}\\ x\times 15.50&=&~~~~~62.00\\ \color{#669900}{\div15.50}&&~\color{#669900}{\div15.50}\\ x&=&4~~~~~ \end{array}$

  So Ben and his siblings could buy $4$ games at Tami's Toys.

  The Ice Bear

  • $\$126.50$ for the game system
  • $\$10.50$ for each game

  $\begin{array}{rcr} x\times 10.50+126.50&=&200.00\\ \color{#669900}{-126.50}&&\color{#669900}{-126.50}\\ x\times 10.50&=&73.50\\ \color{#669900}{\div10.50}&&\color{#669900}{\div10.50}\\ x&=&7~~~~~ \end{array}$

  So Ben and his siblings could buy $7$ games at The Ice Bear.

  Punju's Palace

  • $\$97.50$ for the game system
  • $\$10.25$ for each game

  $\begin{array}{rcr} x\times 10.25+97.50&=&200.00\\ \color{#669900}{-97.50}&&\color{#669900}{-97.50}\\ x\times 10.25&=&102.50\\ \color{#669900}{\div10.25}&&\color{#669900}{\div10.25}\\ x&=&10~~~~~ \end{array}$

  So Ben and his siblings could buy $10$ games at Punju's Palace.

  The Gold Pig

  • $\$145.00$ for the game system
  • $\$5.00$ for each game

  $\begin{array}{rcr} x\times 5.00+145.00&=&200.00\\ \color{#669900}{-145.00}&&\color{#669900}{-145.00}\\ x\times 5.00&=&55.00\\ \color{#669900}{\div5.00}&&\color{#669900}{\div5.00}\\ x&=&11~~~~~ \end{array}$

  So Ben and his siblings could buy $11$ games at The Gold Pig.