Solve Equations Using the Subtraction and Addition Properties of Equality

Determine whether a number is a solution of an equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation — it is the answer to the puzzle.

Can you recognize the solution of $x + 2 = 7$? If you said $5$, you’re right — substituting $5$ for $x$ gives $5 + 2 = 7$, a true statement, so $5$ is a solution.

Example. Determine whether $x = 5$ is a solution of $6x - 17 = 16$.

Substitute $5$ for $x$: $6(5) - 17 = 16$. Multiply: $30 - 17 = 16$. Subtract: $13 \ne 16$. Since the resulting statement is false, $x = 5$ is not a solution.

Example. Determine whether $y = 2$ is a solution of $6y - 4 = 5y - 2$. Here the variable appears on both sides, so substitute $2$ for each $y$: $6(2) - 4 = 5(2) - 2$. Multiply: $12 - 4 = 10 - 2$. Subtract: $8 = 8$, a true statement — so $y = 2$ is a solution.

Solve equations using the Subtraction Property of Equality

The goal in solving an equation is to isolate the variable by itself on one side of the equation. The Subtraction Property of Equality says that subtracting the same quantity from both sides of an equation keeps the two sides equal.

Think of twin brothers, both age $17$. Three years ago, both were $17 - 3 = 14$ — subtracting the same amount from equal quantities keeps them equal.

Example. Solve $x + 8 = 17$. Subtract $8$ from both sides: $x + 8 - 8 = 17 - 8$. Simplify: $x = 9$. Check: $9 + 8 = 17$. ✓

Example. Solve $100 = y + 74$. It doesn’t matter which side the variable is on — subtract $74$ from both sides: $100 - 74 = y + 74 - 74$. Simplify: $26 = y$. Check: $100 = 26 + 74$. ✓

Solve equations using the Addition Property of Equality

Some equations subtract a number from the variable, such as $x - 5 = 8$. To “undo” the subtraction, add the same number to both sides — the Addition Property of Equality.

Back to the twins, both $17$: in ten years, both will still be equal — $17 + 10 = 27$ for each. Adding the same number to both sides keeps them equal.

Example. Solve $x - 5 = 8$. Add $5$ to both sides: $x - 5 + 5 = 8 + 5$. Simplify: $x = 13$. Check: $13 - 5 = 8$. ✓

Example. Solve $27 = a - 16$. Add $16$ to both sides: $27 + 16 = a - 16 + 16$. Simplify: $43 = a$. Check: $27 = 43 - 16$. ✓

Translate word phrases to algebraic equations

An equation has an equal sign between two expressions, so a sentence saying two phrases are equal translates into an equation. Watch for clue words that mean equals: is equal to, is the same as, is, gives, was, will be.

It helps to box the equals word first, then translate each phrase on either side of it.

Example. Translate: “The sum of $6$ and $9$ is $15$.” The word is marks the equal sign: $6 + 9 = 15$.

Example. Translate: “The product of $8$ and $7$ is $56$.” Result: $8 \cdot 7 = 56$.

Example. Translate: “Twice the difference of $x$ and $3$ gives $18$.” Twice means two times, and difference of x and 3 means $x - 3$; the whole difference is doubled, so it needs parentheses: $2(x - 3) = 18$.

Translate to an equation and solve

Now combine both skills: translate a sentence into an equation, then solve it using the properties above.

Example. Translate and solve: “Three more than $x$ is equal to $47$.” Translate: $x + 3 = 47$. Subtract $3$ from both sides: $x + 3 - 3 = 47 - 3$. Simplify: $x = 44$. Check: $44 + 3 = 47$. ✓

Example. Translate and solve: “The difference of $y$ and $14$ is $18$.” Translate: $y - 14 = 18$. Add $14$ to both sides: $y - 14 + 14 = 18 + 14$. Simplify: $y = 32$. Check: $32 - 14 = 18$. ✓

Key terms

solution — a value of a variable that makes a true statement when substituted into an equation. solving an equation — the process of finding a solution. Subtraction Property of Equality — subtracting the same quantity from both sides of an equation keeps the two sides equal. Addition Property of Equality — adding the same quantity to both sides of an equation keeps the two sides equal.

Practice

Determine whether a number is a solution of an equation

Solve equations using the Subtraction Property of Equality

Solve equations using the Addition Property of Equality

Translate word phrases to algebraic equations

Translate to an equation and solve

This section is adapted from Prealgebra 2e, Section 2.3: Solve Equations Using the Subtraction and Addition Properties of Equality by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, replaced the envelope-and-counters manipulative figures with a direct statement of the Subtraction Property, converted practice problems (“Try Its”) into interactive exercises with instant feedback, and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.