4 Writing and Solving Equations

Learn to translate verbal statements into equations, solve one-step and two-step equations while keeping both sides balanced, and check solutions in the original problem.

From words to equations

To solve a word problem with an , name the unknown with a , translate the statement into symbols, solve the , and check that the answer fits the original statement.

Translating words into equations

Common word clues indicate which operation or symbol to use:

Addition: “sum,” “increased by,” or “more than.” For example, $7$ more than $x$ is $x + 7$ .
Subtraction: “difference,” “decreased by,” or “less than.” For example, $5$ less than $x$ is $x - 5$ .
Multiplication: “product,” “times,” or “of.” For example, $3$ times $x$ is $3x$ .
Division: “quotient” or “divided by.” For example, $x$ divided by $4$ is $\frac{x}{4}$ .
Equality: “is,” “equals,” or “is the same as” signals an equals sign.

Order matters in subtraction and division. “ $5$ less than a number” means $x - 5$ , not $5 - x$ . “The quotient of a number and $4$ ” means $\frac{x}{4}$ .

For example, translate “Three times a number, increased by $4$ , is $19$ .” Let $x$ be the number. “Three times a number” gives $3x$ ; increasing that by $4$ gives $3x + 4$ ; and “is $19$ ” gives the :

3x + 4 = 19

Solving one-step equations

An stays balanced when the same operation is applied to both sides. To isolate the , use the operation that undoes the operation applied to it:

Undo addition by subtracting the same amount from both sides.
Undo subtraction by adding the same amount to both sides.
Undo multiplication by dividing both sides by the same nonzero number.
Undo division by multiplying both sides by the same number.

For example, solve $x + 8 = 21$ by subtracting $8$ from both sides:

x + 8 - 8 = 21 - 8

This gives $x = 13$ . Check the solution in the : $13 + 8 = 21$ , so the solution is correct.

For $5x = 35$ , divide both sides by $5$ :

\frac{5x}{5} = \frac{35}{5}

Therefore, $x = 7$ .

Solving two-step equations

A has two operations applied to the . Undo them in reverse order: first undo addition or subtraction, then undo multiplication or division.

To solve $4x + 7 = 31$ :

Subtract $7$ from both sides to get $4x = 24$ .
Divide both sides by $4$ to get $x = 6$ .

Check by substituting into the original : $4(6) + 7 = 24 + 7 = 31$ .

For a word problem, translate the full statement before solving. If a number is multiplied by $3$ , then $5$ is added, and the result is $26$ , let $x$ be the number. The is $3x + 5 = 26$ . Subtract $5$ to get $3x = 21$ , then divide by $3$ to get $x = 7$ . Check: $3(7) + 5 = 26$ . The number is $7$ .

A reliable method and practice

Use this process to solve a word problem:

Identify the unknown and define a for it.
Translate the whole statement into an . Use the equality clue, often “is,” to place the equals sign.
Solve by doing the same operation to both sides and undoing operations in reverse order.
Substitute the answer into the original or statement to check it.
State the answer with appropriate units when the problem gives units.

For practice, write an and solve each problem:

A number decreased by $9$ is $14$ : $x - 9 = 14$ , so $x = 23$ .
Twice a number plus $3$ is $17$ : $2x + 3 = 17$ , so $x = 7$ .
Four times a number is $36$ : $4x = 36$ , so $x = 9$ .

Translate phrases carefully, especially for subtraction and division, and use an equals sign to express equality. Keep both sides balanced while solving, then check the answer by substituting it into the original .