Eight children were given some candy. Then six different children were given the same unknown amount of candy. Next, two children were that same unknown amount of candy plus three additional pieces of candy. The total number of pieces of candy given out was thirty eight. What is the unknown amount of candy?

In this concept, you will learn to solve equations with the distributive property and combining.

**Distributive Property and Combining Like Terms**

To solve some multi-step equations you will need to use the distributive property and combine like terms. When this happens, you will see that there is more than one term with the same variable or there is more than one number in the equation. You always want to combine everything that you can before moving on to solving the equation.

Let‘s apply this to the following situation.

Solve for “\begin{align*}m\end{align*}” in the following equation.

\begin{align*}6(1+2m)-3m=24\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 6 and then add those products.

Next, combine like terms (

and ) on the left side of the equation.

Then, solve as you would solve any two-step equation. Subtract 6 from both sides of the equation.

\begin{align*}\begin{array}{rcl} 6+9m &=& 24\\ 6-6+9m &=& 24-6 \\ 9m &=& 18 \end{array}\end{align*}

Then, divide both sides of the equation by 9 to solve for

.\begin{align*}\begin{array}{rcl} 9m &=& 18\\ \frac{9m}{9}&=& \frac{18}{9} \\ m &=& 2 \end{array}\end{align*}

The answer is 2.

Here is another example.

Solve for “\begin{align*}b\end{align*}” in the following equation.

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by -4 and then add those products.

Next, add the like terms on the left side of the equation. To add those like terms, \begin{align*}5b\end{align*}, you will need to use what you know about adding integers.

and\begin{align*}\begin{array}{rcl} -8+(-12b)+5b &=& 13\\ -8+(-12b+5b) &=& 13 \\ -8+(-7b) &=& 13 \end{array}\end{align*}

Then, solve as you would solve any two-step equation. Since -8 is added to

, you can subtract -8 from both sides of the equation to solve it.\begin{align*}\begin{array}{rcl} -8+(-7b) &=& 13\\ -8-(-8)+(-7b)&=& 13-(-8) \\ (-8+8)+(-7b) &=& 13+8 \\ -7b &=& 21 \end{array}\end{align*}

Then, divide both sides of the equation by -7.

\begin{align*}\begin{array}{rcl} -7b &=& 21\\ \frac{-7b}{-7}&=& \frac{21}{-7} \\ b &=& -3 \end{array}\end{align*}

The answer is -3.

### Examples

#### Example 1

Earlier, you were given a problem about eight children who were given some candy. You will let “\begin{align*}c\end{align*}” represent the unknown amount of candy given.

Six of the eight children were given the same unknown amount of candy \begin{align*}(2(c+3))\end{align*}. The total number of pieces of candy given out was thirty eight.

and two of the children were that same unknown amount of candy plus three additional pieces of candyFirst, write an equation.

First, apply the distributive property to the left side of the equation.

Next, add the like terms on the left side of the equation.

Then, subtract 6 from both sides.

\begin{align*}\begin{array}{rcl}
8c+6=38 \\
8c+6-6=38-6 \\
8c = 32
\end{array}\end{align*}

Then, divide both sides of the equation by 8.

The answer is 4.

Therefore, the unknown amount of candy is 4 pieces. Six of the children got 4 pieces of candy and two of the children received 7 pieces of candy.

#### Example 2

Solve for “

in the following equation.\begin{align*}-5x+3(x+1)-4x=45\end{align*}

First, apply the distributive property to the left side of the equation.

Next, add the like terms on the left side of the equation.

Then, subtract 3 from both sides.

Then, divide both sides of the equation by -6.

\begin{align*}\begin{array}{rcl} -6x &=& 42\\ \frac{-6x}{-6}&=& \frac{42}{-6} \\ x &=& -7 \end{array}\end{align*}

The answer is -7.

#### Example 3

Solve for “

” in the following equation.

First, apply the distributive property to the left side of the equation.

Next, add the like terms on the left side of the equation.

Then, subtract 22 from both sides.

Then, divide both sides of the equation by 9.

The answer is 4.

#### Example 4

Solve for “

” in the following equation.

First, apply the distributive property to the left side of the equation.

\begin{align*}\begin{array}{rcl} 6y+3(y-4) &=& 33\\ 6y+(3 \times y)+(3x \times -4)&=& 33 \\ 6y+3y+(-12) &=& 33 \end{array}\end{align*}

Next, add the like terms on the left side of the equation.

\begin{align*}\begin{array}{rcl} 6y+3y+(-12) &=& 33\\ (6y+3y)+(-12)&=& 33 \\ 9y-12 &=&33 \end{array}\end{align*}

Then, add 12 to both sides.

\begin{align*}\begin{array}{rcl} 9y-12 &=& 33\\ 9y-12+12&=& 33+12 \\ 9y &=& 45 \end{array}\end{align*}

Then, divide both sides of the equation by 9.

\begin{align*}\begin{array}{rcl} 9y &=& 45\\ \frac{9y}{9}&=& \frac{45}{9} \\ y &=& 5 \end{array}\end{align*}

The answer is 5.

#### Example 5

Solve for “

” in the following equation.

First, apply the distributive property to the left side of the equation.

Next, add the like terms on the left side of the equation.

Then, subtract 21 from both sides.

Then, divide both sides of the equation by 19.

The answer is 1.

**Review**

Distribute and combine like terms and then solve each equation.

- \begin{align*}9y + 3(y− 6) = 30\end{align*}
- \begin{align*}3(y− 1) + 2(y + 3) = 13\end{align*}
- \begin{align*}6(x + 2) − 4x + 6 = 36\end{align*}
- \begin{align*}−9(x + 3) + 4x = −2\end{align*}
- \begin{align*}4(a + 2) − 9 = 11\end{align*}
- \begin{align*}−8(y + 2) − 16 = 16\end{align*}