3. Evaluating Expressions by Substitution

To evaluate an expression, replace the variable with a specific number and simplify. Substitution must be careful, especially for negative numbers.

Example:
\[ 4x-7\quad\text{when }x=3. \]
Replace \(x\) with 3:
\[ 4(3)-7=12-7=5. \]

For a negative input, use parentheses:
\[ x^2+3x\quad\text{when }x=-2. \]
Substitute:
\[ (-2)^2+3(-2)=4-6=-2. \]

Expressions with two variables work the same way. If \(a=5\) and \(b=-3\), then:
\[ 2a-b=2(5)-(-3)=10+3=13. \]

If an expression contains a fraction, substitute first and then simplify:
\[ \frac{3x+6}{2}\quad\text{when }x=4 \]
\[ \frac{3(4)+6}{2}=\frac{18}{2}=9. \]

Common misconception: writing \(3(-2)^2\) when the expression is \(3x^2\) and thinking the 3 is squared too. Only the variable part is squared unless parentheses say otherwise.

Academic habit: put every substituted negative number in parentheses. This prevents sign and exponent mistakes.