Introduction

Evaluating an algebraic expression means replacing variables with numbers and then carrying out the arithmetic.
This article assumes you already know what variables and constants are.

Our goal is to make substitution feel natural and intuitive, even if you’re new to algebra.

What Is an Algebraic Expression?

An algebraic expression is a combination of:

• variables (like $x$, $y$, $t$)
• numbers
• arithmetic operations (addition, subtraction, multiplication, division)

Examples:

• $3x + 2$
• $5(a - 1)$
• $2m + 3n - 7$

These are not equations — they don’t have an equals sign. They’re simply instructions for computation once you know the values of the variables.

Why Substitution Matters

Substitution lets you:

• compute the value of an expression for specific inputs
• test whether a value “works” in a formula
• understand how formulas behave in real situations (distance, cost, temperature, etc.)

It’s one of the most common skills in all of mathematics.

How to Substitute Values

When you substitute, you follow three simple steps:

1. Replace each variable with the given number.
   Example: replace $x$ with $4$ in $3x + 2$.
2. Use parentheses when substituting.
   This avoids mistakes, especially with negatives.
   Example: $5 - 2x$ becomes $5 - 2(4)$.
3. Carry out the arithmetic carefully.
   Follow the usual order of operations:
   • parentheses
   • multiplication/division
   • addition/subtraction

Worked Examples

Example 1: Evaluate $3x + 2$ when $x = 4$

• Substitute: $3(4) + 2$
• Multiply: $12 + 2$
• Final answer: 14

Example 2: Evaluate $5 - 2x$ when $x = -3$

• Substitute: $5 - 2(-3)$
• Multiply: $5 + 6$
• Final answer: 11

Example 3: Evaluate $2a + 3b$ when $a = 1$ and $b = 5$

• Substitute: $2(1) + 3(5)$
• Compute: $2 + 15$
• Final answer: 17

Example 4: Evaluate $t^2 - 4$ when $t = 6$

• Substitute: $(6)^2 - 4$
• Compute: $36 - 4$
• Final answer: 32

Common Mistakes to Avoid

• Forgetting parentheses when substituting negative numbers
  • Example: writing $-3^2$ instead of $(-3)^2$
• Doing addition before multiplication
  • Example: computing $3x + 2$ as $3(x + 2)$ (incorrect)
• Dropping signs
  • Especially when subtracting negative numbers

Calculator

Exercises

1. Evaluate $4x + 1$ when $x = 3$.

   Solution

   $4x + 1$ with $x = 3$ $$4(3) + 1 = 12 + 1 = \mathbf{13}$$
2. Evaluate $2y - 5$ when $y = -2$.

   Solution

   $2y - 5$ with $y = -2$ $$2(-2) - 5 = -4 - 5 = \mathbf{-9}$$
3. Evaluate $3a + 4b$ when $a = 2$ and $b = 1$.

   Solution

   $3a + 4b$ with $a = 2$, $b = 1$ $$3(2) + 4(1) = 6 + 4 = \mathbf{10}$$
4. Evaluate $t^2 + t$ when $t = 5$.

   Solution

   $t^2 + t$ with $t = 5$ $$5^2 + 5 = 25 + 5 = \mathbf{30}$$
5. Evaluate $6 - 2m$ when $m = -4$.

   Solution

   $6 - 2m$ with $m = -4$ $$6 - 2(-4) = 6 + 8 = \mathbf{14}$$
6. Evaluate $3(x - 1)$ when $x = 7$.

   Solution

   $3(x - 1)$ with $x = 7$ $$3(7 - 1) = 3(6) = \mathbf{18}$$
7. Evaluate $2p^2 - p$ when $p = 3$.

   Solution

   $2p^2 - p$ with $p = 3$ $$2(3^2) - 3 = 2(9) - 3 = 18 - 3 = \mathbf{15}$$
8. Evaluate $5 - (2q + 1)$ when $q = 2$.

   Solution

   $5 - (2q + 1)$ with $q = 2$ $$5 - (2(2) + 1) = 5 - (4 + 1) = 5 - 5 = \mathbf{0}$$