How do I evaluate an expression when a variable has a given value?

Evaluating an expression means finding its numerical value after replacing each variable with the number you’re given. The reliable routine is: substitute first, then simplify using the order of operations. [S1]

Try this sequence:

1. Copy the expression and replace every instance of the variable.
2. Put parentheses around a substituted negative number.
3. Simplify powers, then multiplication or division, then addition or subtraction.
4. Check that you used the given value everywhere.

For example, if a = 4, evaluate 3a + 2: 3(4) + 2 = 12 + 2 = 14.

The variable is a placeholder for the given number in this calculation. Don’t solve for the variable; it already has a value.

After substitution, treat the expression as a normal arithmetic problem. Work out powers before multiplication, and multiplication before addition. Parentheses show which number is being substituted and help keep the expression’s meaning clear. [S1]

Worked example: let x = 3. Evaluate 2x² + 5.

1. Substitute 3 for each x: 2(3)² + 5.
2. Evaluate the power: 2(9) + 5.
3. Multiply: 18 + 5.
4. Add: 23.

A useful distinction: 2x² means 2 times x squared. With x = 3, that is 2(3²) = 18, not (2 × 3)². Keep the original expression’s structure as you substitute.

Put the negative value in parentheses wherever you substitute it, especially when it is raised to a power. This makes clear that the whole negative number is replacing the variable. [S1]

Let y = -2. Evaluate y² + 3y:

1. Substitute: (-2)² + 3(-2).
2. Square the grouped value: 4 + 3(-2).
3. Multiply: 4 - 6.
4. Add: -2.

Be careful not to change the expression while substituting. In y², the variable is squared, so write (-2)², which equals 4. In -y², the minus sign is outside the square: substituting gives -((-2)²), or -4. The parentheses help you read which part is being squared; they don’t mean you should ignore the original operations.

Yes. First work each one without looking at the answers; then compare your steps, not just your final number.

Practice set:

1. If m = 5, evaluate 2m + 7.
2. If p = -3, evaluate p² + 2p.
3. If n = -2, evaluate 3(n + 1)².

Answers:

1. 2(5) + 7 = 10 + 7 = 17.
2. (-3)² + 2(-3) = 9 - 6 = 3.
3. 3((-2) + 1)² = 3(-1)² = 3.

If yours differs, check that every variable was replaced, negative values were grouped, and powers were evaluated before the operations outside them. For the third problem, simplify inside the parentheses before squaring; the grouping is part of the expression.