Combine Like Terms and Distribute to Solve Equations | Grade 8 Math | LevelKite

You already know how to solve a two-step equation such as $3x + 5 = 20$ . An equation says that two expressions have the same value. A solution is a value of the variable that makes the equation true. Here $x = 5$ is a solution, because $3(5) + 5 = 15 + 5 = 20$ .

The equations in this lesson start out messier. The variable may appear more than once, as in $7x + 5 - 3x = 29$ . Or it may sit inside parentheses, as in $3(x + 6) = 30$ . Your goal is to tidy each one into a two-step equation without changing its solutions.

In this lesson, $x$ can stand for any real number, which means any number on the number line. We call the numbers a variable may stand for its permitted values.

The first way to tidy an equation is to rewrite one side.

Two expressions are equivalent when they have the same value for every permitted value of the variable. Suppose you replace one side of an equation with an equivalent expression. The value of that side does not change for any $x$ , so the new equation is true for exactly the same values of $x$ as the old one.

Combining like terms is one of these rewrites. Like terms have the same variable raised to the same power, such as $7x$ and $-3x$ . Plain numbers, such as $5$ and $9$ , are like terms too. Seven of a number minus three of the same number leaves four of it, so $7x - 3x = 4x$ for every value of $x$ .

The second way is to do the same operation to both sides.

When two sides are equal, they stay equal if you add the same number to both or subtract the same number from both. They also stay equal if you multiply or divide both by the same number, as long as that number is not zero. You can reverse each of these steps, so the new equation has exactly the same solutions as the old one. No solution is lost, and no new one appears.

Zero is the exception. You can never divide by zero. Multiplying both sides by zero turns any equation into $0 = 0$ , which tells you nothing about $x$ .

Worked example

Combine like terms, then solve

Solve $7x + 5 - 3x = 29$ .

Find the like terms on the left side: 7x and 3x have the same variable, x. The 3x is subtracted, and subtracting 3x is the same as adding negative 3x, so the left side is a sum. You can change the order of a sum without changing its value. Put the like terms next to each other and keep the minus sign with 3x.
$7x - 3x + 5 = 29$
Combine the like terms. Seven x's minus three x's leaves four x's. The new left side is equivalent to the old one, so the equation has the same solutions.
$(7 - 3)x + 5 = 29 \quad\text{so}\quad 4x + 5 = 29$
Now you have a two-step equation. Subtract 5 from both sides to undo the added 5. Both sides change by the same amount, so they stay equal.
$4x + 5 - 5 = 29 - 5 \quad\text{so}\quad 4x = 24$
Divide both sides by 4 to undo the multiplication by 4. This is valid because 4 is not zero.
$\frac{4x}{4} = \frac{24}{4} \quad\text{so}\quad x = 6$
Check in the original equation by replacing x with 6. The left side equals 29, the same as the right side, so 6 makes the equation true.
$7(6) + 5 - 3(6) = 42 + 5 - 18 = 29$

Answer

The solution is $x = 6$ .

Now look at parentheses. In $3(x + 6)$ , the $3$ multiplies the whole group inside the parentheses. So the expression means three copies of $x + 6$ :

$(x + 6) + (x + 6) + (x + 6) = 3x + 18$

Three copies of the group contain three $x$ 's and three $6$ 's. So $3(x + 6)$ and $3x + 18$ are equivalent, and you may replace one with the other in an equation without changing its solutions.

The distributive property states this pattern for all numbers $a$ , $b$ and $c$ . When you use it to remove parentheses, you are distributing.

a(b + c) = ab + ac

Worked example

Distribute, then solve

Solve $3(x + 6) = 30$ .

Distribute the 3 to both terms inside the parentheses: 3 times x and 3 times 6. The new left side is equivalent to the old one, so the equation has the same solutions.
$3 \cdot x + 3 \cdot 6 = 30 \quad\text{so}\quad 3x + 18 = 30$
Subtract 18 from both sides to undo the added 18. Both sides change by the same amount, so they stay equal.
$3x + 18 - 18 = 30 - 18 \quad\text{so}\quad 3x = 12$
Divide both sides by 3 to undo the multiplication by 3. This is valid because 3 is not zero.
$\frac{3x}{3} = \frac{12}{3} \quad\text{so}\quad x = 4$
Check in the original equation by replacing x with 4. Add inside the parentheses first. The left side equals 30, the same as the right side, so 4 makes the equation true.
$3(4 + 6) = 3(10) = 30$

Answer

The solution is $x = 4$ .

Which rewrite matches 3(x + 6) at each value of x: 3x + 18 or 3x + 6?

x	3(x + 6)	3x + 18	3x + 6
$-2$	$12$	$12$	$0$
$0$	$18$	$18$	$6$
$1$	$21$	$21$	$9$
$4$	$30$	$30$	$18$
$8$	$42$	$42$	$30$

Multiply every term inside the parentheses. A common mistake is to multiply only the first term and rewrite $3(x + 6) = 30$ as $3x + 6 = 30$ . That gives $3x = 24$ , so $x = 8$ . But the check in the original equation fails: $3(8 + 6) = 3(14) = 42$ , not $30$ .

Why does it fail? $3(x + 6)$ means three copies of the whole group, and each copy contains a $6$ . In the table, $3x + 6$ is $12$ less than $3(x + 6)$ in every row. That gap is the two missing $6$ 's. The expressions are not equivalent, so the swap makes a different equation with a different solution. The column for $3x + 18$ matches the column for $3(x + 6)$ in every row, and the distributive property guarantees the match for every value of $x$ .

Key concept

Simplify each side, then undo

To solve an equation that has parentheses or several like terms on one side, simplify before you undo anything.

Distribute to remove parentheses.
Combine like terms that are on the same side.
Undo addition or subtraction, then multiplication or division. Do the same operation on both sides, and multiply or divide only by a number that is not zero.
Check your value in the original equation.

Steps 1 and 2 replace a side with an equivalent expression. Step 3 changes both sides in the same way. So every line has exactly the same solutions as the first line.

Worked example

Distribute a negative number, then combine

Solve $-2(x - 5) + 6x = 2$ .

Distribute negative 2 to both terms inside the parentheses. Think of x minus 5 as x plus negative 5. Negative 2 times x is negative 2x. Negative 2 times negative 5 is positive 10, because the product of two negative numbers is positive. The new left side is equivalent to the old one.
$-2 \cdot x + (-2)(-5) + 6x = 2 \quad\text{so}\quad -2x + 10 + 6x = 2$
Combine the like terms, negative 2x and 6x. Put them next to each other, which does not change the value of the sum. Negative 2 plus 6 is 4, so they combine to 4x.
$-2x + 6x + 10 = 2 \quad\text{so}\quad 4x + 10 = 2$
Subtract 10 from both sides, which keeps them equal.
$4x + 10 - 10 = 2 - 10 \quad\text{so}\quad 4x = -8$
Divide both sides by 4, which is not zero. A negative number divided by a positive number is negative.
$\frac{4x}{4} = \frac{-8}{4} \quad\text{so}\quad x = -2$
Check in the original equation by replacing x with negative 2. The left side equals 2, the same as the right side, so negative 2 makes the equation true.
$-2(-2 - 5) + 6(-2) = -2(-7) - 12 = 14 - 12 = 2$

Answer

The solution is $x = -2$ .

Takeaway. When an equation has parentheses or several like terms, simplify each side first: distribute, then combine like terms. Each rewrite gives an equivalent expression. Then undo operations by doing the same operation on both sides, never multiplying or dividing by zero. Because every step is one of those two kinds, each line has the same solutions as the first, and a check in the original equation confirms your answer.

Next, you will solve longer multi-step equations, including ones where the numbers multiplying the variable are fractions or decimals. You will also justify each step and find and correct errors in solutions.