Evaluating Algebraic Expressions
Substitute values for variables and simplify expressions using the order of operations. Include multi-variable expressions and formulas used in science and finance contexts.
A lesson of Algebra 1, in Foundations of Algebra, about 30 minutes.
By the end of this lesson, you will be able to evaluate any algebraic expression by substituting given values and simplifying in the correct order, including expressions with several variables, negative numbers, exponents and fraction bars.
An Expression Is a Machine You Put Numbers Into
An expression like $3x^2 - 5x + 2$ does not have a value on its own. It has a value once you say what $x$ is.
Feed it $4$ and it hands back $30$. Feed it $0$ and it hands back $2$. Feed it $-1$ and it hands back $10$. The expression is the machine, the number you feed it is the input, and the number that comes out is the value of the expression at that input.
Putting a number in and turning the handle is called evaluating.
You will do it constantly: checking whether a solution to an equation actually works, using a formula in science, reading a graph, predicting from a model. And nearly every mistake anyone makes doing it comes from the same place, which is doing the operations in the wrong order.
Two Steps, and They Never Change
- Substitute. Replace each variable with the given value, in brackets.
- Simplify. Work through the order of operations until one number is left.
Why the brackets matter. Substituting a negative number without them changes what the expression says. Take $-x^2$ with $x = 3$:
$$-(3)^2 = -9 \quad \checkmark$$
$$-3^2 \text{ looks like it might be } (-3)^2 = 9 \quad \times$$
The brackets cost nothing and remove the ambiguity, so put them in every time, including when the number is positive. A habit you only use half the time is not a habit.
The Order of Operations
| Priority | Operation | Notes |
|---|---|---|
| 1 | Parentheses | Innermost first |
| 2 | Exponents | Powers and roots |
| 3 | Multiplication and Division | Equal priority, left to right |
| 4 | Addition and Subtraction | Equal priority, left to right |
The two lines that catch people are the last two. Multiplication does not outrank division, and addition does not outrank subtraction. Inside each of those pairs you go left to right, in reading order.
$$8 - 5 + 2 = 3 + 2 = 5$$
Not $8 - 7 = 1$. The subtraction happens first because it is on the left.
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