Evaluating an algebraic expression means replacing each variable with a given number,
then simplifying the numerical expression using the correct order of operations.
1. What does “evaluate” mean?
An expression such as
\[
\begin{aligned}
3x^2-5y+2
\end{aligned}
\]
does not have one fixed value until values for \(x\) and \(y\) are given.
If \(x=-2\) and \(y=4\), we substitute those values into the expression.
2. Substitution
Substitute \(x=-2\) and \(y=4\):
\[
\begin{aligned}
3x^2-5y+2
&=
3(-2)^2-5(4)+2.
\end{aligned}
\]
Parentheses around negative substituted values are important.
Without parentheses, the square may be interpreted incorrectly.
3. Follow the order of operations
Use the standard order:
- Parentheses
- Exponents
- Multiplication and division from left to right
- Addition and subtraction from left to right
For the sample:
\[
\begin{aligned}
3(-2)^2-5(4)+2
&=
3(4)-20+2\\
&=
12-20+2\\
&=
-6.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
3x^2-5y+2=-6
\quad\text{when }x=-2,\ y=4.
}
\end{aligned}
\]
4. Exponents with negative values
Be careful when a negative number is raised to a power:
\[
\begin{aligned}
(-2)^2&=4,\\
-2^2&=-4.
\end{aligned}
\]
The first expression squares the whole number \(-2\).
The second expression applies the negative sign after squaring \(2\).
5. Fractions in algebraic evaluation
Fractions should be kept exact during the calculation when possible.
For example:
\[
\begin{aligned}
\frac{x}{2}+\frac{3y}{4}
\quad\text{when }x=5,\ y=6.
\end{aligned}
\]
Substitute:
\[
\begin{aligned}
\frac{5}{2}+\frac{3(6)}{4}
&=
\frac{5}{2}+\frac{18}{4}\\
&=
\frac{5}{2}+\frac{9}{2}\\
&=
\frac{14}{2}\\
&=
7.
\end{aligned}
\]
6. Multiple variables
Expressions may contain more than one variable.
For example:
\[
\begin{aligned}
2a+3b-c.
\end{aligned}
\]
If \(a=5\), \(b=-2\), and \(c=7\), then:
\[
\begin{aligned}
2(5)+3(-2)-7
&=
10-6-7\\
&=
-3.
\end{aligned}
\]
7. Implicit multiplication
In algebra, writing \(3x\) means \(3\cdot x\).
Similarly:
\[
\begin{aligned}
xy&=x\cdot y,\\
2(x+1)&=2\cdot(x+1).
\end{aligned}
\]
During evaluation, these implied multiplications must still be performed.
8. Parentheses change the value
Compare:
\[
\begin{aligned}
2x+1
\quad\text{and}\quad
2(x+1).
\end{aligned}
\]
If \(x=3\):
\[
\begin{aligned}
2x+1
&=
2(3)+1
=
7,\\
2(x+1)
&=
2(3+1)
=
8.
\end{aligned}
\]
The same numbers appear, but the grouping is different.
9. Checking an answer
A good check is to rewrite the expression after substitution and make sure each operation is performed in the correct order.
For the sample:
\[
\begin{aligned}
3(-2)^2-5(4)+2
&=
3(4)-20+2\\
&=
12-20+2\\
&=
-6.
\end{aligned}
\]
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Forgetting parentheses around negative substituted values.
- Evaluating multiplication before exponents.
- Replacing only one occurrence of a variable instead of every occurrence.
- Confusing \(2x+1\) with \(2(x+1)\).
- Dropping negative signs during substitution.
- Converting fractions to rounded decimals too early.
Key idea: substitute carefully first, then simplify the numerical expression using the order of operations.
