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Lesson 4: Substitution & evaluating expressions

# Evaluating expressions with one variable

A mixture of explanations, examples, and practice problems to have you evaluating expressions with one variable in no time!

## How to evaluate an expression with one variable

Let's say we want to evaluate the expression a, plus, 4. Well, first we need to know the value of the variable a. For example, to evaluate the expression when start color #11accd, a, equals, 1, end color #11accd, we just replace start color #11accd, a, end color #11accd with start color #11accd, 1, end color #11accd:
\begin{aligned} &\blueD a + 4 \\\\ =&\blueD1 + 4~~~~~~~~\gray{\text{Replace }\blueD{a} \text{ with } \blueD{1}\text{.}} \\\\ =&5 \end{aligned}
So, the expression a, plus, 4 equals 5 when a, equals, 1.
We can just as easily evaluate a, plus, 4 when start color #11accd, a, equals, 5, end color #11accd:
\begin{aligned} &\blueD a + 4 \\\\ =&\blueD5 + 4~~~~~~~~\gray{\text{Replace }\blueD{a} \text{ with } \blueD{5}\text{.}} \\\\ =&9 \end{aligned}
So, the expression a, plus, 4 equals 9 when a, equals, 5.

## Evaluating an expression with multiplication

You might be asked to "Evaluate 3, x when x, equals, 5."
Notice how the number 3 is right next to the variable x in the expression 3, x. This means "3 times x". The reason we do this is because the old way of showing multiplication with the symbol times looks confusingly similar to the variable x.
Okay, so now let's solve the problem:
\begin{aligned} &3\blueD x \\\\ =& 3 \cdot \blueD5~~~~~~~~\text{Replace }\blueD{x} \text{ with } \blueD{5}\text{.} \\\\ =&15 \end{aligned}
So, the expression 3, x equals 15 when x, equals, 5.

### New ways to show multiplication

Hold on a second! Did you notice that we wrote "3 times start color #11accd, 5, end color #11accd" as 3, dot, start color #11accd, 5, end color #11accd instead of as 3, times, start color #11accd, 5, end color #11accd? Using a dot instead of the symbol times is another new way of showing multiplication:
3, dot, start color #11accd, 5, end color #11accd, equals, 15
Parentheses can also be used to show multiplication:
3, left parenthesis, start color #11accd, 5, end color #11accd, right parenthesis, equals, 15
Let's summarize the new ways of showing multiplication that we learned.
Old wayNew way
With a variable3, times, x3, x
Without variable3, times, 53, dot, 5 or 3, left parenthesis, 5, right parenthesis

## Evaluating expressions where order of operations matter

For more complex expressions, we'll have to be sure to pay close attention to order of operations. Let's take a look at an example:
Evaluate 5, plus, 3, e when start color #11accd, e, equals, 4, end color #11accd.
\begin{aligned} &5+3\blueD e \\\\ =&5 + 3 \cdot \blueD 4~~~~~~~~\gray{\text{Replace }\blueD{e} \text{ with } \blueD{4}\text{.}} \\\\ =&5 + 12 ~~~~~~~~\text{\gray{Multiply first (order of operations)}} \\\\ =&17 \end{aligned}
So, the expression 5, plus, 3, e equals 17 when e, equals, 4.
Notice how we had to be careful to think about order of operations when evaluating. A common wrong answer is start color #e84d39, 32, end color #e84d39, which comes from first adding 5 and 3 to get 8 then multiplying 8 by 4 to get start color #e84d39, 32, end color #e84d39.

## Let's practice!

Problem 1
• Current
Evaluate the expression 9, minus, z when z, equals, 4.

## Challenge problems

Challenge problem 1
• Current
Evaluate e, dot, e, minus, 5, e when e, equals, 5.