Solving Multi-Step Equations with Variables on Both Sides

Solving Multi-Step Equations with Variables on Both Sides exercise

• Determine how many Grey Borings Emilio needs to exchange to buy one more Rainbow Blinkey.

  Hints

  Imagine an equation as a scale in balance.

  Whenever you add or subtract a value or term from one side of an equation, you have to do it on the other side of the equation as well. Otherwise, it doesn't stay in balance.

  If one Rainbow Blinkey costs $x$, for example, three Rainbow Blinkeys cost $3x$.

  Solution

  Emilio would like to know how many Grey Borings he needs to exchange so he can buy one more Rainbow Blinkey. First, we assign the cost of one Rainbow Blinkey to the variable $x$.

  Joanna buys $5$ Rainbow Blinkeys and only one Grey Boring. So Joanna spends $5x+1$. Emilio buys $10$ Grey Borings and $2$ Rainbow Blinkeys that cost $2x+10$. Since they spend the same amount of money, we set our two expressions equal to each other.

  $5x+1=2x+10$

  Imagine a scale in balance. Each operation has to be performed on both sides of the scale:

  1. subtract $1$: $5x=2x+9$
  2. subtract $2x$: $3x=9$
  3. divide by $3$: $x=3$

  The cost of one Rainbow Blinkey is $3$ times the cost of one Grey Boring. In other words, Emilio needs to exchange $3$ Grey Borings for $1$ Rainbow Blinkey.

• Decide which steps are needed to solve the equation $3\times (2x-5)+10=4x+9$.

  Hints

  Imagine an equation as a scale in balance. After we combine like terms, we have to Isolate the Variable. Each operation we perform, we have to perform on both sides of the equation.

  Use Opposite Operations:

  • Addition $\longleftrightarrow$ Subtraction, and vice versa
  • Multiplication $\longleftrightarrow$ Division, and vice versa

  Remember to always solve equations step-by-step.

  Solution

  Here, we'll see how we can solve a more complicated equation using different mathematical rules by going step-by-step. The equation we want to solve is $3\times(2x-5)+10=4x+9$.

  1. First, we use the Distributive Property to get rid of the parentheses.
  2. That leaves us with $6x-15+10=4x+9$.
  3. Then, we combine like terms.
  4. Doing this gives us $6x-5=4x+9$.
  5. Next, we add $5$ on both sides, which gives us $6x=4x+14$.
  6. Subtracting $4x$ from both sides will leave us with $2x=14$.
  7. Finally, dividing by $2$ gives us the solution $x=7$.

  Notice how the last three steps all follow the same concept: Opposite Operations.

• Find the right equations for the fish on the scale.

  Hints

  If you buy one goldfish for the price of $x$, $4$ goldfish will cost $4x$.

  You have to add the costs of the different types of fish.

  Solution

  Let's start with the scale on the left:

  • We see $2$ goldfish and $6$ Rainbow Blinkeys on the left side of the scale. This gives us the expression $2x+6$.
  • On the right side of the scale are $4$ goldfish and $2$ Rainbow Blinkeys, leading to the expression $4x+2$.
  • Because Joanna and Emilio spend the same amount of money, we set our two expressions equal to each other, giving us $2x+6=4x+2$.

  Now let's have a look at the scale in the middle:

  • We see $3$ goldfish and $2$ Grey Borings on the left side of the scale. This gives us the term $3x+2$.
  • On the right side of the scale are $2$ goldfish and $5$ Grey Borings. So we get $2x+5$.
  • Because Joanna and Emilio spend the same amount of money, our final equation is $3x+2=2x+5$.

  At last, we examine the scale on the right:

  • We see $4$ goldfish on the left side of the scale, $4x$.
  • On the right side of the scale are $1$ goldfish and $9$ Grey Borings. This gives us the term $x+9$.
  • Because Joanna and Emilio spend the same amount of money, our final equation is $4x=x+9$.

• Decide when the numbers of inhabitants match.

  Hints

  If $50$ people move away after one year, then $100$ people move away after two years.

  You can solve an equation by using opposite operations:

  • The opposite operation for addition is subtraction, and vice versa.
  • The opposite operation for multiplication is division, and vice versa.

  When the number of inhabitants in both villages match, there will be $2,600$ people in each village.

  Solution

  In general, we solve algebraic problems with variables on both sides by following these steps:

  1. We assign the variable $x$ to the unknown number. In this problem, it's the number of years.
  2. Next, we define expressions according to the story:

  • village Addition $2000+150x$
  • village Subtraction $3000-100x$

  3. To find out how many years until the villages will have the same numbers of inhabitants, we write the equation $2000+150x=3000-100x$.

  How do we solve the equation? To solve, isolate the variable $x$.

  • First, we arrange the equation so that all constants are on one side of the equal sign and the variables on the other. In order to do this, we use opposite operations to move the terms.
  • Subtracting $2000$ on both sides gives us $150x=1000-100x$.
  • Now we add $100x$ on both sides to get $250x=1000$.
  • Finally, we divide by $250$, which gives us $x=4$.

  In $4$ years, $2600$ people will live in each village.

• Evaluate how we can solve an equation.

  Hints

  Look at the scale above. What do you see?

  In the middle of the scale you can imagine an equal sign.

  Both sides of an equal sign represent the same value.

  If you only put a weight on the left side of a scale, the scale will be imbalanced. Since you want the scale to be balanced, you have to put the same weight on the right side as well.

  Solution

  Solving an equation can be thought of as a scale in balance.

  In the middle of an equation there is an equal sign. On both sides we find equal terms.

  Back to the scale: each operation has to be performed on both sides. Otherwise the scale will not be in balance. For example, in order to solve the equation $3x=9$, we have to divide both sides by $3$. That will leave us with $x=3$.

  Remember the rules concerning opposite operations:

  • the opposite operation of addition is subtraction, and vice versa
  • the opposite operation of multiplication is division, and vice versa

• Solve the following equations.

  Hints

  Use the Distributive Property to simplify, if necessary. For example: $3\times(x+2)=3x+6$.

  Combine like terms on one or both sides of the equation.

  Isolate the variable $x$ by using Opposite Operations.

  Solution

  Because you've learned already a lot, we should practice, but this time let's evaluate equations that are more complicated. The concept always stays the same. Step by step, if it's necessary:

  1. First we use the Distributive Property.
  2. Next, we Combine Like Terms.
  3. Then, we move the constants to the right side and the variables to the left, or vice versa.
  4. Finally, we divide on both sides to get the solution.

  $\begin{array}{rcrcl} 2\times(x-1)+2&=&4\times(x+1)-6\\ 2\times x-2+2&=&4\times x+4-6&&|\text{ Distributive Property}\\ 2x&=&4x-2&&|\text{ Combine Like Terms}\\ \color{#669900}{-4x}&&\color{#669900}{-4x}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ -2x&=&-2\\ \color{#669900}{\div (-2)}&&\color{#669900}{\div (-2)}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&1 \end{array}$

  Using the same steps, we solve two more equations:

  $\begin{array}{rcrcl} 5\times(x+2)-20&=&3\times(x+1)-5\\ 5\times x+10-20&=&3\times x+3-5&&|\text{ Distributive Property}\\ 5x-10&=&3x-2&&|\text{ Combine Like Terms}\\ \color{#669900}{+10}&&\color{#669900}{+10}&&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ 5x&=&3x+8\\ \color{#669900}{-3x}&&\color{#669900}{-3x}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 2x&=&8\\ \color{#669900}{\div2}&&\color{#669900}{\div2}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&4 \end{array}$

  ... and ...

  $\begin{array}{rcrcl} 12\times(x-1)&=&5x+23\\ 12x-12&=&5x+23&&|\text{ Distributive Property}\\ \color{#669900}{+12}&&\color{#669900}{+12}&&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ 12x&=&5x+35\\ \color{#669900}{-5x}&&\color{#669900}{-5x}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 7x&=&35\\ \color{#669900}{\div7}&&\color{#669900}{\div7}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&5 \end{array}$