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Simplifying Expressions with Variables
> ... Math > The Order of Operations and Variables > Simplifying Expressions with Variables

Introduction

In this lesson, you will learn about using variables. It is important to remember that simplifying expressions with variables is dependent on knowing what the variables are.

It is important to keep in mind the order of operations when solving equations that involve variables. PEMDAS: Parentheses, exponents, multiplication, division, addition, and subtraction.

This video illustrates the lesson material below. Watching the video is optional.

Simplifying Variable Expressions

Example 1
Solve: \(4n\)

You can’t simplify this example, because you don’t know what n is equal to. But if it said:

Solve: \(4n\)
\(n=3\)

You can simplify this expression. \(4n\) becomes \(4(3)\), which also means \4\times3\. The answer to this equation is 12.

Example 2
Solve: \(4n\)
\(n=5\)

Simplify this statement by placing 5 in the n’s place: \(4(5)\) or \(4\times5\). This equals 20.

Example 3
Solve: \(4n + \frac{6}{n}\)
\(n=3\)

Solve this by rewriting the expression to say: \(4(3) + \frac{6}{3}\). In order to solve this, you must remember the order of operations. Multiplication and division come before addition, so the first thing you will do is multiply \(4\times3\) and divide \(6\div3\).

The expression should now look like this: \(12+2\).

The last thing you'll do is add the two numbers together, which equals 14.

Remember, if the value of n were to change, the answer would be different.

Example 4
Solve: \(3m - 2w\)
This expression has two different variables:
\(m = 3\) and \(w = 4\)

Substitute the value of each variable into the expression: \(3(3) - 2(4)\). Using order of operations, \(3\times3\) and \(2\times4\).

Simplifying the expression looks like: \(9 - 8\). Subtract 8 from 9 and the final answer is 1.

Example 5
Solve: \(y\cdot m-2\)
\(y=3\)
\(m=10\)

Substitute the y and m values into the equation:

\(3\cdot10-2\)

Begin with the multiplication: \(3\times10=30\).

The expression will look like this after you multiply: \(30-2\).

After subtracting 2 from 30, you'll find the answer: 28.

Variables in Formulas

Simplifying expressions with variables is helpful when you are given a formula for a certain type of problem. A formula is something that looks like \(pt = a\). This is a formula for finding the amount when you know the percent and the total. You learned about this formula in a previous lesson on percentages.

In this case, \(p\) represents the percent and \(t\) represents the total. If you were told that the percent was \(75\%\) and the total was \(20\), you could find that the amount was \(15\). This is done by putting the values in for the variables in the formula. \(pt\) becomes \(\lgroup.75\ngroup\lgroup20\ngroup\) which is equal to \(15\). The answer is \(a = 15\).

Things to Remember


  • The order of operations (PEMDAS).
  • You can only simplify or solve an expression with a variable if you know the value of the variable.

Practice Problems

Simplify the following expression to find its value:
1. Given that \(d = 3\):
\(7d = ?\) (
Solution
x
Solution: 21
Details:
\(7d = 7 × 3 = 21\)

A number next to a variable implies multiplication.

\(7d\)

Replace the \(d\) in the expression with 3, as given in the math problem. Include a multiplication symbol of your choice.

\(7 × {\color{green}3}\)

Finally, multiply \(7 × 3\).

The answer is \(21\).
)2. Given that \(m = 20\):
\(m − 12 = ?\) (
Solution
x
Solution: 8
Details:
\(m − 12 = 20 − 12 = 8\)
)3. Given that \(x = 2\):
\(6x + 3 = ?\) (
Solution
x
Solution: 15
Details:
\(6x + 3 = 6 · 2 + 3 = 15\)

In this problem, the x is the variable.

\(6 \lgroup x \rgroup +3\)

Replace the variable \(x\) with 2 because \(x = 2\) is given in the math problem. Use parentheses or a dot to show multiplication and to distinguish the symbol for multiplication from the \(x\) variable.

\(6\lgroup2\rgroup+ 3\)

Now use the order of operations to solve the problem. First, multiply \(6\) and \(2\).

\(12+ 3\)

Lastly, add \(12 + 3\).

The answer is 15.
 |
Video Solution
x

 | Transcript)As you will explore later, the area of a rectangle can be found by multiplying the length of the rectangle by its width \({\text{(L × W)}}\). Use this information to answer the following two questions:4. Find the area of a rectangular-shaped floor, where the length is \(L = 3\) meters and the width is \(W = 4\) meters. (
Video Solution
x
Solution: 12
Details:
\({\text{L × W}} = 3 × 4 = 12\)

As you will study later in the course, the units in this answer are square meters. So, you can say the room has a size of 12 square meters. At this time, you will not worry about the units.


 | Transcript)5. Find the area of a rectangular-shaped computer monitor, where the length is \(L = 31\; cm\) and the width is \(W = 17\; cm\). (
Solution
x
Solution: 527
Details:
\(L \times W = 31 × 17 = 527\)

As you will study later, the units in this answer are square centimeters (\(cm^{2})\). So, you can say the monitor has a size of 527 square centimeters. At this time, you will not worry about the units.
)
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