Evaluating Algebraic Expressions

Substitution

Meaning: Substitution means replacing a variable with a number.

Definition: To evaluate an expression, substitute the given value and simplify.

If $x = 4$, then 3$x + 2$ becomes 3$(4) + 2$.

• Replace every copy of the variable.
• Use brackets for negative values.
• Simplify after substitution.

Order of Operations

Meaning: Calculations must be done in the correct order.

Definition: Order of operations decides which operation is done first.

In $2 + 3$x when $x = 4$, multiply first: $2 + 12 = 14$.

• Brackets first.
• Then powers.
• Then multiplication and division.
• Then addition and subtraction.

Expressions with Two Variables

Meaning: Some expressions use more than one variable.

Definition: Each variable must be replaced by its own given value.

If $a = 2$ and $b = 5$, then 3$a + b = 3$$(2) + 5 = 11$.

• Do not use the same value for every variable unless the question says so.

Negative Values

Meaning: Negative substitutions need brackets.

Definition: A negative value should be substituted in parentheses.

If x = -3, then $x^{2}$ means $(-3)^2 = 9$.

Easy

Question: Find the value of 3$x + 5$ when $x = 4$.

1. Substitute $x = 4$.
2. 3$x + 5 = 3$$(4) + 5$.
3. 3$(4) = 12$.
4. $12 + 5 = 17$.

Medium

Question: Find 2$a + 3$b when $a = 5$ and $b = 2$.

1. Substitute $a = 5$ and $b = 2$.
2. 2$a + 3$$b = 2$$(5) + 3$(2).
3. 2$(5) = 10$ and 3$(2) = 6$.
4. The value is 16.

Exam-level

Question: Evaluate $x^{2} - 4$$x + 1$ when x = -3.

1. Use brackets for the negative value.
2. $x^{2} - 4$$x + 1 = (-3)^2 - 4$$(-3) + 1$.
3. $(-3)^2 = 9$.
4. -4$(-3) = 12$.
5. $9 + 12 + 1 = 22$.