Evaluating expressions

Once you have an algebraic expression and the value of each letter, you simply substitute and apply BODMAS to compute the numeric result.

Idea

To evaluate $3x + 5$ at $x = 4$:

1. Replace $x$ by $4$: $3(4) + 5$.
2. Use BODMAS: multiplication first $3 \times 4 = 12$, then addition $12 + 5 = 17$.

So $3x + 5 = 17$ when $x = 4$.

A few cautions:

• When you substitute a negative number, use brackets: $3(-2) + 5 = -6 + 5 = -1$. Without brackets you may forget the sign.
• For an expression with several letters like $2x + 3y$, you need a value for each letter.
• For squared or cubed letters: $x^2$ at $x = 5$ is $5^2 = 25$, not $5 \times 2$.

Substitution lets us see how a single expression takes on many values as the variable changes. The expression $2x + 3$ at $x = 0, 1, 2, 3$ gives $3, 5, 7, 9$. The pattern (each value is $2$ more than the last) is the rate of change, and the constant $3$ is the value at $x = 0$.

This idea , plug in a value, get a result , is the foundation of formulas, functions, and graphs in higher classes.

Worked examples

Example 1. Evaluate $4y - 3$ at $y = 5$.

$4(5) - 3 = 20 - 3 = 17$.

Example 2. Evaluate $2x + 3y$ at $x = 4$, $y = 1$.

$2(4) + 3(1) = 8 + 3 = 11$.

Example 3. Evaluate $x^2 + 2x$ at $x = -3$.

$x^2 = (-3)^2 = 9$ and $2x = 2(-3) = -6$. Sum: $9 + (-6) = 3$.

Example 4. A taxi charges $12x + 50$ rupees for $x$ km. Find the fare for (i) $5$ km, (ii) $20$ km.

(i) $12(5) + 50 = 60 + 50 = \rupee 110$. (ii) $12(20) + 50 = 240 + 50 = \rupee 290$.

Try it yourself

1. Evaluate $5n - 2$ at $n = 3$.
2. Evaluate $a + b$ at $a = 7$, $b = -2$.
3. Evaluate $2x^2$ at $x = 4$.
4. Evaluate $3(x + 5)$ at $x = 2$.
5. The perimeter of a square is $4s$. Find it when $s = 9$ cm.
6. Evaluate $5x - 3y$ at $x = 4$, $y = 2$.
7. Evaluate $\tfrac{x + y}{2}$ at $x = 6$, $y = 10$.
8. The cost expression is $C = 20n + 100$. Find $C$ for $n = 0, 5, 10$.

Activity

Make a table for $y = 2x + 1$ at $x = 0, 1, 2, 3, 4, 5$. Plot $(x, y)$ on graph paper. Notice they form a straight line , that's a sneak peek of next year's algebra.