Evaluating Expressions: Substituting a Value

Replace the letter with the number

Evaluating an expression means substituting a value for each variable and working out the result.

Evaluate $3n + 5$ at $n = 4$:

$$3(4) + 5 = 12 + 5 = 17$$

The expression is the rule; evaluating it answers one particular case of that rule.

$n$	$3n + 5$
$0$	$5$
$4$	$17$
$10$	$35$

Substitute with brackets

Write the value inside brackets where the letter was:

$$3n \;\rightarrow\; 3(4) \qquad n^2 \;\rightarrow\; (4)^2 \qquad 5 - n \;\rightarrow\; 5 - (4)$$

With a positive value the brackets look unnecessary. With a negative one they decide the answer.

$$3n \text{ at } n = -2: \qquad 3(-2) = -6$$

Writing it without brackets gives $3-2 = 1$, which is a different question entirely — the minus has turned from part of the number into an operation.

Then follow the order of operations

Once the numbers are in, it is ordinary arithmetic: brackets, exponents, multiplication and division left to right, then addition and subtraction left to right.

Evaluate $2x^2 + 3$ at $x = 5$:

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

The exponent comes before the multiplication:

$$2(25) + 3 = 50 + 3 = 53$$

Multiplying first would give $10^2 + 3 = 103$, which answers a different question. The $2$ multiplies $x^2$, not $x$.

Negatives and squares

This pair is worth separating carefully.

Expression	At $x = 3$	Result
$x^2$	$(3)^2$	$9$
$-x^2$	$-(3)^2$	$-9$
$(-x)^2$	$(-3)^2$	$9$

In $-x^2$ the minus sits outside the squaring, so only the $3$ is squared and the minus survives. In $(-x)^2$ the brackets take the negative inside, so it is squared too and two negatives make a positive.

Evaluating $x^2$ at $x = -3$ gives $(-3)^2 = 9$, which is why the brackets in the substitution matter so much.

Why this is the skill everything else rests on

Substitution is how a general rule produces a specific answer, and almost every later technique is an application of it.

• Checking a solution. Solving $3n + 5 = 20$ gives $n = 5$. Substituting it back gives $3(5) + 5 = 20$, and the check is finished.
• Building a table of values. Plotting a line means evaluating its equation at several $x$ values and reading off the $y$s.
• Using a formula. $A = \pi r^2$ becomes a number only once a radius is substituted.

That first use is the one worth adopting as a habit. A solved equation can always be verified by putting the answer back into the original, and an answer that fails the check is caught before it is written down. It turns a guess into something you know.

Worked examples

Common mistakes

Practice problems