Table of Contents
Evaluating algebraic expressions means answering the question: “What number does this expression represent when I choose specific values for the variables?”
In this chapter, we assume you already know what algebraic expressions and variables are. Here, we focus on how to correctly substitute numbers for variables and simplify.
The basic idea: Substitute, then simplify
An algebraic expression is a combination of numbers, variables, and operation symbols (like $+$, $-$, $\times$, $\div$, exponents).
To evaluate an expression:
- Substitute the given value(s) for each variable.
- Follow the order of operations to simplify to a single number.
The standard order of operations can be remembered as:
- Parentheses
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
You may already know this from arithmetic; here you apply it after substituting.
Example:
Evaluate $3x + 5$ when $x = 4$.
- Substitute $x = 4$:
$x + 5 \to 3(4) + 5$$ - Simplify:
$(4) + 5 = 12 + 5 = 17$$
So, when $x = 4$, the value of $3x + 5$ is $17$.
Using substitution carefully
Substitution means replacing each variable with its specific value, everywhere it appears.
Example:
Evaluate $2x - x^2$ when $x = 3$.
- Substitute:
$x - x^2 \to 2(3) - (3)^2$$ - Simplify step by step:
$(3) = 6,$$
$$(3)^2 = 9,$$
so
$ - 9 = -3.$$
A common mistake is to miss a place where the variable appears, or to forget the exponent.
Using parentheses when substituting
Whenever you substitute a number (especially a negative number) into an expression, it is safer to put that number in parentheses. This avoids confusion about signs and exponents.
Example with a negative number:
Evaluate $x^2 - 4x$ when $x = -3$.
- Substitute with parentheses:
$$x^2 - 4x \to (-3)^2 - 4(-3)$$ - Simplify:
$$(-3)^2 = 9,$$
$(-3) = -12,$$
so
$ - (-12) = 9 + 12 = 21.$$
Notice the difference between $(-3)^2$ and $-3^2$ (the second one means $-(3^2)$). Parentheses keep the meaning clear.
Evaluating expressions with more than one variable
Sometimes an expression uses several variables. You must substitute each one with its given value.
Example:
Evaluate $2x + 3y$ when $x = 1$ and $y = 4$.
- Substitute:
$x + 3y \to 2(1) + 3(4)$$ - Simplify:
$(1) = 2,\quad 3(4) = 12,$$
so
$ + 12 = 14.$$
Another example:
Evaluate $ab - 2b^2$ when $a = 5$ and $b = -1$.
- Substitute with parentheses:
$$ab - 2b^2 \to (5)(-1) - 2(-1)^2$$ - Simplify:
$$(5)(-1) = -5,$$
$$(-1)^2 = 1,$$
so
$$-5 - 2(1) = -5 - 2 = -7.$$
Using the order of operations in expressions
After substitution, you handle the expression using the usual order of operations. The expression may have:
- Parentheses
- Exponents
- Multiplication and division
- Addition and subtraction
Example:
Evaluate $3(2x - 1)^2$ when $x = 2$.
- Substitute:
$(2x - 1)^2 \to 3(2(2) - 1)^2.$$ - Inside parentheses:
$(2) = 4,$$
so
$(4 - 1)^2 = 3(3)^2.$$ - Exponent:
$$(3)^2 = 9,$$
so
$(9).$$ - Multiply:
$(9) = 27.$$
You should not, for example, multiply $3 \times 2$ first and ignore the parentheses.
Evaluating expressions with fractions and division
Expressions may contain fractions or division signs. After substituting, you simplify as with arithmetic.
Example:
Evaluate $\dfrac{x + 2}{3}$ when $x = 7$.
- Substitute:
$$\dfrac{x + 2}{3} \to \dfrac{7 + 2}{3}.$$ - Simplify numerator:
$ + 2 = 9,$$
so
$$\dfrac{9}{3} = 3.$$
Example with two variables:
Evaluate $\dfrac{2a - b}{5}$ when $a = 4$ and $b = 9$.
- Substitute:
$$\dfrac{2a - b}{5} \to \dfrac{2(4) - 9}{5}.$$ - Simplify numerator:
$(4) = 8,$$
$ - 9 = -1,$$
so
$$\dfrac{-1}{5} = -\dfrac{1}{5}.$$
Evaluating expressions with exponents
When the expression has exponents, you must be very precise about what is being raised to a power.
Example:
Evaluate $2x^2 + 1$ when $x = 3$.
- Substitute:
$x^2 + 1 \to 2(3)^2 + 1.$$ - Exponent first:
$$(3)^2 = 9,$$
so
$(9) + 1 = 18 + 1 = 19.$$
Example with a negative value:
Evaluate $x^3 - x$ when $x = -2$.
- Substitute with parentheses:
$$x^3 - x \to (-2)^3 - (-2).$$ - Exponent:
$$(-2)^3 = -8,$$
so
$$-8 - (-2) = -8 + 2 = -6.$$
Always keep negative substituted values inside parentheses to avoid errors.
Evaluating expressions with absolute value
If an expression has absolute value bars $|\,\cdot\,|$, remember that $|n|$ is the distance from $n$ to $0$ on the number line, always non-negative.
Example:
Evaluate $|x - 5|$ when $x = 2$.
- Substitute:
$$|x - 5| \to |2 - 5|.$$ - Simplify inside:
$ - 5 = -3,$$
so
$$|-3| = 3.$$
Another example:
Evaluate $2|y| - 1$ when $y = -4$.
- Substitute:
$|y| - 1 \to 2|-4| - 1.$$ - Absolute value:
$$|-4| = 4,$$
so
$(4) - 1 = 8 - 1 = 7.$$
Checking your work
After evaluating, it is helpful to:
- Repeat the substitution step silently and see if you replaced every variable correctly.
- Check the order in which you simplified: parentheses → exponents → multiplication/division → addition/subtraction.
- Estimate whether your final answer is reasonable (for example, by roughly thinking about the size and sign of the terms).
Example check:
For $3x - 4$ with $x = -2$, you get:
$$3(-2) - 4 = -6 - 4 = -10.$$
Quick check: $3x$ is $3$ times a negative number, so it should be negative. You then subtract $4$, making it more negative; $-10$ makes sense.
Evaluating expressions from words
Sometimes you are given a description in words and a value for the variable. You may first write the expression, then evaluate.
Example:
“The expression is 5 more than twice a number $n$. Evaluate it when $n = 6$.”
- “Twice a number $n$” is $2n$.
- “5 more than” means add 5: $2n + 5$.
- Substitute $n = 6$:
$(6) + 5 = 12 + 5 = 17.$$
Separating “write the expression” from “evaluate the expression” helps you keep your steps clear.
Common mistakes to avoid
- Forgetting parentheses when substituting negatives:
- Wrong: $x^2$ with $x = -3$ as $-3^2 = -9$.
- Correct: $(-3)^2 = 9$.
- Changing the expression while evaluating:
- Do not turn $3x^2$ into $(3x)^2$; they are different.
- Ignoring the order of operations:
- For $2 + 3x$ with $x = 4$, do not do $(2+3)\cdot 4$.
- Correct: $2 + 3(4) = 2 + 12 = 14$.
Summary
- To evaluate an expression, replace each variable with its given value and simplify using the usual order of operations.
- Use parentheses when substituting, especially with negative numbers or exponents.
- Pay attention to every part of the expression: exponents, parentheses, fractions, and absolute values.
- Evaluate step by step, and check your final answer for reasonableness.