Evaluate, Simplify, and Translate Expressions

Evaluate algebraic expressions

In the last section, we simplified expressions using the order of operations. Here we evaluate expressions — find the value of an expression when the variable is replaced by a given number. To evaluate, substitute the number for the variable and then simplify using the order of operations.

Example. Evaluate $x + 7$ when (a) $x = 3$ and (b) $x = 12$.

For (a), substitute $3$ for $x$: $3 + 7 = 10$. For (b), substitute $12$ for $x$: $12 + 7 = 19$. Notice the two parts give different results, because the value of an expression depends on the value used for the variable.

Example. Evaluate $9x - 2$ when (a) $x = 5$ and (b) $x = 1$. Remember that $9x$ means $9$ times $x$. For (a): $9 \cdot 5 - 2 = 45 - 2 = 43$. For (b): $9(1) - 2 = 9 - 2 = 7$. Both the dot and the parentheses tell us to multiply.

When an expression contains a variable with an exponent, substitute carefully.

Example. Evaluate $x^2$ when $x = 10$. Substitute $10$ for $x$: $10^2 = 10 \cdot 10 = 100$.

Example. Evaluate $2^x$ when $x = 5$. Here the variable is the exponent: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$.

An expression can contain more than one variable, requiring more than one substitution.

Example. Evaluate $3x + 4y - 6$ when $x = 10$ and $y = 2$. Substitute: $3(10) + 4(2) - 6$. Multiply: $30 + 8 - 6$. Add and subtract left to right: $32$.

Example. Evaluate $2x^2 + 3x + 8$ when $x = 4$. Be careful: $2x^2$ means $2 \cdot x \cdot x$, which is different from $(2x)^2$. Substitute $4$ for each $x$: $2(4)^2 + 3(4) + 8$. Simplify the exponent: $2(16) + 3(4) + 8$. Multiply: $32 + 12 + 8$. Add: $52$.

Identify terms, coefficients, and like terms

Algebraic expressions are built from terms. A term is a constant or the product of a constant and one or more variables. Examples of terms are $7$, $y$, $5x^2$, $9a$, and $13xy$.

The constant that multiplies the variable(s) in a term is called the coefficient. Think of it as the number in front of the variable. The coefficient of $3x$ is $3$. When we write $x$ by itself, the coefficient is $1$, since $x = 1 \cdot x$.

An expression may have several terms added or subtracted; we include the operation before a term with it. For example, $3x^2 + 4x^2 + 5y + 3$ has the terms $3x^2$, $4x^2$, $5y$, and $3$.

Example. Identify the terms of $9b + 15x^2 + a + 6$ and each coefficient. The four terms are $9b$, $15x^2$, $a$, and $6$. Their coefficients are $9$, $15$, $1$ (no number written means $1$), and $6$ (the coefficient of a constant is the constant itself).

Some terms share the same variables and exponents. Like terms are terms that are either constants or have the same variables raised to the same powers. Among the terms $5x$, $7$, $n^2$, $4$, $3x$, $9n^2$: the constants $7$ and $4$ are like terms; $5x$ and $3x$ are like terms; and $n^2$ and $9n^2$ are like terms.

Simplify expressions by combining like terms

We can simplify an expression by combining like terms. What does $3x + 6x$ simplify to? If you have $3$ of something and add $6$ more of the same thing, you have $9$ of them, so $3x + 6x = 9x$. We add the coefficients and keep the same variable.

Example. Simplify $3x + 7 + 4x + 5$. The like terms are $3x$ and $4x$, and the constants $7$ and $5$. Rearranged: $3x + 4x + 7 + 5$. Combine: $7x + 12$.

When a term has a negative coefficient, the procedure is the same — you subtract instead of add.

Example. Simplify $7x^2 + 8x - x^2 - 4x$. The like terms are $7x^2$ and $-x^2$, and $8x$ and $-4x$. Rearranged: $7x^2 - x^2 + 8x - 4x$. Combine: $6x^2 + 4x$. Since $6x^2$ and $4x$ are not like terms, this is in simplest form.

Translate word phrases to algebraic expressions

In the last section we translated expressions into words; now we reverse the process. Watch for the words of and and to find the numbers being operated on.

Example. Translate each phrase into an algebraic expression: (a) the difference of $20$ and $4$; (b) the quotient of $10x$ and $3$.

For (a), difference means subtraction: $20 - 4$. For (b), quotient means division: $10x \div 3$, which can also be written $\tfrac{10x}{3}$.

Two phrases need special care. More than means “added to,” and less than means “subtracted from” — so the order is reversed from how the words are read.

Example. Translate each phrase: (a) eight more than $y$; (b) seven less than $9z$. For (a), “more than” means added to $y$: $y + 8$. For (b), “less than” means subtracted from $9z$: $9z - 7$.

Parentheses matter when a phrase combines operations.

Example. Translate: (a) five times the sum of $m$ and $n$; (b) the sum of five times $m$ and $n$. In (a) we multiply $5$ by the whole sum, so we need parentheses: $5(m + n)$. In (b) we add $n$ to five times $m$: $5m + n$. The parentheses change the result.

Key terms

evaluate — to find the value of an expression by substituting a given number for the variable and simplifying. term — a constant or the product of a constant and one or more variables. coefficient — the constant that multiplies the variable(s) in a term. like terms — terms that are constants, or that have the same variables raised to the same powers. combining like terms — simplifying by adding the coefficients of like terms.

Practice

Evaluate algebraic expressions

Identify terms, coefficients, and like terms

Simplify expressions by combining like terms

Translate word phrases to algebraic expressions

This section is adapted from Prealgebra 2e, Section 2.2: Evaluate, Simplify, and Translate Expressions 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, recreated tables in accessible Markdown, converted practice problems (“Try Its”) into interactive exercises with instant feedback, and adapted selected end-of-section exercises into the interactive Practice block.