Evaluating Algebraic Expressions

Once you know what variables and expressions are, the next step is to evaluate them — replace every variable with a specific number and calculate the result. This process is called substitution, and it is the single most-used skill in applied mathematics. Every time a nurse plugs patient data into a dosage formula or an electrician calculates wire ampacity from a table, they are evaluating an algebraic expression.

What Does “Evaluate” Mean?

To evaluate an expression means to find its numerical value after replacing each variable with a given number. The steps are always the same:

1. Write the original expression.
2. Replace each variable with the given value (use parentheses around the substituted number).
3. Follow the order of operations (PEMDAS) to simplify.

Using parentheses during substitution prevents sign errors, especially with negative numbers.

Evaluating with One Variable

Example 1: Evaluate $3x + 7$ when $x = 4$

Step 1 — Substitute:

$3(4) + 7$

Step 2 — Multiply:

$12 + 7$

Step 3 — Add:

$19$

Answer: 19

Example 2: Evaluate $x^2 - 5x + 6$ when $x = 3$

Step 1 — Substitute:

$(3)^2 - 5(3) + 6$

Step 2 — Exponent:

$9 - 5(3) + 6$

Step 3 — Multiply:

$9 - 15 + 6$

Step 4 — Left to right:

$-6 + 6 = 0$

Answer: 0

Example 3: Evaluate $2x^2 + 1$ when $x = -3$

Step 1 — Substitute (parentheses are critical here):

$2(-3)^2 + 1$

Step 2 — Exponent first: $(-3)^2 = 9$

$2(9) + 1$

Step 3 — Multiply, then add:

$18 + 1 = 19$

Answer: 19

Notice that $(-3)^2 = 9$ (positive), not $-9$. The parentheses around $-3$ mean the entire number is squared. Without parentheses, $-3^2 = -(3^2) = -9$.

Evaluating with Multiple Variables

Example 4: Evaluate $2a + 3b$ when $a = 5$ and $b = -2$

Step 1 — Substitute:

$2(5) + 3(-2)$

Step 2 — Multiply each term:

$10 + (-6)$

Step 3 — Add:

$4$

Answer: 4

Example 5: Evaluate $\frac{x + y}{x - y}$ when $x = 10$ and $y = 4$

Step 1 — Substitute:

$\frac{(10) + (4)}{(10) - (4)}$

Step 2 — Simplify numerator and denominator:

$\frac{14}{6}$

Step 3 — Reduce the fraction:

$\frac{7}{3}$

Answer: $\dfrac{7}{3}$ (or approximately $2.33$)

Evaluating Formulas

Formulas are just expressions with specific variable names. The evaluation process is identical.

Example 6: Area of a Trapezoid

The area formula for a trapezoid is:

$A = \frac{1}{2}(b_1 + b_2)h$

Find the area when $b_1 = 8$ cm, $b_2 = 12$ cm, and $h = 5$ cm.

Step 1 — Substitute:

$A = \frac{1}{2}(8 + 12)(5)$

Step 2 — Parentheses:

$A = \frac{1}{2}(20)(5)$

Step 3 — Multiply left to right:

$A = 10 \times 5 = 50$

Answer: $A = 50$ cm$^2$

Example 7: Temperature Conversion

Convert Celsius to Fahrenheit using:

$F = \frac{9}{5}C + 32$

Find $F$ when $C = 20$.

Step 1 — Substitute:

$F = \frac{9}{5}(20) + 32$

Step 2 — Multiply:

$F = 36 + 32$

Step 3 — Add:

$F = 68$

Answer: 20 degrees Celsius equals 68 degrees Fahrenheit.

Real-World Application: Electrician — Power Formula

An electrician uses the electrical power formula:

$P = I^2 \times R$

where $P$ is power in watts, $I$ is current in amps, and $R$ is resistance in ohms. A 15-amp circuit with 8 ohms of resistance uses:

Step 1 — Substitute:

$P = (15)^2 \times 8$

Step 2 — Exponent:

$P = 225 \times 8$

Step 3 — Multiply:

$P = 1{,}800$

Answer: The circuit draws 1,800 watts. This tells the electrician whether the circuit can handle the load — a standard 15-amp, 120-volt circuit provides 1,800 watts, so this is at full capacity.

Real-World Application: Nursing — Dosage Calculation

A nurse must calculate the number of tablets to administer using the formula:

$\text{Tablets} = \frac{D}{H} \times Q$

where $D$ is the desired dose, $H$ is the dose on hand, and $Q$ is the quantity per tablet. If the order is for $D = 500$ mg and each tablet contains $H = 250$ mg with $Q = 1$ tablet:

$\text{Tablets} = \frac{500}{250} \times 1 = 2 \times 1 = 2$

Answer: Administer 2 tablets. Accurate substitution into dosage formulas is critical in clinical practice — an error could harm a patient.

Common Mistakes to Avoid

1. Dropping parentheses around negative numbers. Always write $(-3)^2$ instead of $-3^2$. The first equals $9$; the second equals $-9$.

2. Forgetting order of operations after substitution. Once you replace the variable, the expression is purely numeric — use PEMDAS as usual. Do not just multiply left to right without checking for exponents or parentheses.

3. Substituting into the wrong variable. When an expression has $x$ and $y$, double-check which value goes where. Writing them in parentheses helps: replace $x$ with $(5)$ and $y$ with $(-2)$.

4. Skipping the parentheses around a negative substitution. If $x = -4$, write $(-4)$ wherever $x$ appears. This prevents sign errors in every operation that follows.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Evaluating an expression means substituting given values for every variable and simplifying
• Always use parentheses around the substituted value, especially for negatives
• After substitution, follow PEMDAS exactly as you would with any numeric expression
• Formulas are expressions with named variables — evaluate them the same way
• In trades and healthcare, evaluating formulas accurately is a daily, safety-critical task
• When working with multiple variables, double-check that each value is substituted into the correct position

Return to Pre-Algebra for more topics in this section.