Two-Step Equations: Parentheses, Fractions, Decimals

Read $3(x + 4) = 27$ as a sentence: 3 equal groups, each worth $x + 4$, make $27$ in all. The parentheses tell you that the $3$ multiplies the whole group, not just the $x$.

That meaning already shows you how to solve it. If 3 equal groups make $27$, then one group is worth $27 \div 3 = 9$. So $x + 4 = 9$, and $x$ is $4$ less than $9$, which is $5$.

Notice the order. To build the left side, you start with $x$, add $4$, and multiply by $3$ last. To solve, you work backward. First you undo the multiplying: divide both sides by $3$, which you can do because $3$ is not zero. Then you undo the adding: subtract $4$ from both sides.

There is a second valid route: remove the parentheses first. The distributive property says that multiplying a sum by a number gives the same result as multiplying each part of the sum by that number and then adding the results. Here you distribute the $3$ like this: $3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12$.

Two expressions are equivalent only when they have the same value for every value of the variable. The distributive property makes $3(x + 4)$ and $3x + 12$ equivalent. Rewriting the left side this way changes how it looks, not its value, so the equation keeps the same solution: $3x + 12 = 27$. That is the form you already know how to solve.

The table compares both routes for $3(x + 4) = 27$. Both are valid, so use the one that makes the arithmetic easier for you.

Now let the number that multiplies $x$ be a fraction or a decimal. The number that multiplies a variable is its coefficient, like the $\tfrac{3}{4}$ in $\tfrac{3}{4}x$. The plan stays the same; only the arithmetic changes.

Take $\tfrac{3}{4}x = 9$, which says that three-fourths of $x$ is $9$. To undo multiplying by $\tfrac{3}{4}$, divide both sides by $\tfrac{3}{4}$. Dividing by a nonzero number gives the same result as multiplying by its reciprocal, the number you multiply it by to get $1$. You find the reciprocal of a nonzero fraction by swapping its top and bottom numbers, so the reciprocal of $\tfrac{3}{4}$ is $\tfrac{4}{3}$.

Multiply both sides by $\tfrac{4}{3}$. On the left, $\tfrac{4}{3} \cdot \tfrac{3}{4} = \tfrac{12}{12} = 1$, so you get $1x$, which is $x$. On the right, $9 \cdot \tfrac{4}{3} = \tfrac{36}{3} = 12$. So $x = 12$, and it makes the original equation true: $\tfrac{3}{4} \cdot 12 = \tfrac{36}{4} = 9$.

Takeaway. Parentheses, fractions and decimals change how an equation looks, not how you solve it. Look at what happens to $x$, then undo those operations in reverse order. Do the same operation to both sides, and never divide by zero.

For $p(x + q) = r$ with $p \neq 0$, divide both sides by $p$ and then subtract $q$ from both sides, or first rewrite $p(x + q)$ as the equivalent expression $px + pq$. To undo a nonzero fraction or decimal coefficient, divide both sides by it, which gives the same result as multiplying both sides by its reciprocal. Always check your value in the original equation.

Next, you will solve equations in which you first combine like terms, such as adding $2x$ and $5x$ to get $7x$, and you will solve word problems with two-step equations.