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## Algebra (all content)

### Unit 1: Lesson 2

Introduction to variables

# 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
Evaluate the expression 9, minus, z when z, equals, 4.

## Challenge problems

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

## Want to join the conversation?

• I still don't understand the challenge problems e⋅e−5ee, dot, e, minus, 5, e when e=5e=5e, equals, 5.What does this mean? •   so, e(dot)e - 5e, the dot symbol represents multiplication. And 5e is another way to represent multiplication so, the expression actually looks like this: 5x5 - 5x5 which equals to 0 because 5x5 = 25, and 5x5 = 25, and subtracting 25 with 25 = 0.

e(dot)e - 5e
5x5 - 5x5 = 25 - 25
25 - 25 = 0
• If 9-8/x = ? and x=4 what do I do with the 8/x? Do I multiply add divide or subtract? • If you don't know what y is, then you won't get an exact answer • Well you don't need to know what y is to solve the question. For example you have:6t=48. (if something is written like 6t, that always means you have to multiply, or any variable). So we have: 6t=48, we can divide both sides by 6 (we are dividing because you always do the inverse of what you are trying to solve, the opposite of multiplication is division)

we have to divide both sides by 6

6t=48, 6/6 cancels out, 48/6=8. So now the equation would look like this: t=8, and now we know what t is we can verify it. 6t=48 or 6x8=48.

And that's the beauty of Algebra!

If you need more help just tell me.
-Intellectual Genius
• Why do we have to do multiplication and division first? Why to use these "order of operation"? • Here is why; "Because exponents are simply repeated multiplication, they are performed before multiplication. ... Thus the order: parentheses first, exponents second, mutiplication and division third, and save the lowest level operations of addition and subtraction for last." Personally I believe order of operation is way to balance the equation as well. :)
• so u saying if i put 5 + 2b and be equal to 5 the answer would be 15 • what is it asking me when it says Evaluate 2y2y when y = 6y=6 • •  • Start out with the variable. What is d worth? What do you need to do with that variable in the sentence? You should see that d is worth 4, and we need to now add that in the sentence.

The sentence should now look like this -
8/4 + 3

Now, solve it like you would usually do it. 8/4 is equal to 2 (We need to do it first since it is the second in the Order of Operations), and then add 3 to that 2. You should get an answer of 5 if you did all that calculation correctly.

Hope this helped!
• 