In Unit 2, we solved simple equations and learned about inverse operations. In this Lesson, we will put it all together and learn to solve multi-step equations where we will simplify like terms and perform various inverse operations to be able to isolate the variable. Let’s work out some examples.
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Example 1
Simplify and solve the equation
In order to isolate the variable on the left side, you want to move to the other side of the equation. With the inverse operation of subtraction—that is, addition—add 8 to both sides of the equation.
Next, to eliminate 3 in the denominator, multiply by 3 on both sides.
Your solution is 42.
Check your work.
The solution is correct.
Check
According to the order of operations, you need to first multiply and divide. However, it is simpler to apply inverse operations of addition and subtraction first when you have a variable on only one side of the equation, as in Example 1.
However, when you have more than one fraction in a problem, you want to eliminate the denominator first. An important fact to remember is that when you multiply or divide one term in an equation by a value, you must multiply all terms—on both sides of the equation—by the same value.
Example 2
Solve for the variable in the equation
In this example, since two of the three terms are fractions, you want to eliminate the denominator first. Recall the lowest common multiple (LCM) skill from Unit 1. Here the LCM is 4, which is used to eliminate the denominators in this equation. Apply the inverse operation of division by multiplying each term in the equation by 4, and then simplify.
To isolate the variable, apply the inverse of subtraction and add 10 to each side of the equation.
Lastly, divide by 24 on each side.
The solution is
Check your work.
Your solution is correct.
Check
As you get more practice, you will learn that there are multiple ways to solve equations correctly. Although some methods save time, there is no one best method. It is more a matter of preference.
Example 3
Solve for x in the equation
This equation looks like a proportion, and we can solve it like one. First, simplify to and then cross multiply to eliminate a denominator in the equation.
Next, use the distributive property to simplify.
Use the inverse operation of addition to subtract 6 from both sides.
Lastly, use the inverse operation of multiplication and divide by 2 on both sides.
Your solution is 11.
Check your work.
Your solution is correct.
Check
Example 4
Solve for the variable in the equation
This example is similar to Example 3, except that the variable is in the denominator.
First, cross multiply to eliminate the denominators in the equation.
Next, use the distributive property to simplify.
Use the inverse operation of subtraction and add 15 to both sides of the equation.
Lastly, use the inverse operation of multiplication and divide both sides by 6 to reduce the coefficient of the variable to 1.
Move the variable to the left side.
The solution is equal to 10.
Check your work.
The solution is correct.
Check
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Answer the following questions.
1) Simplify and solve the equation.
2) Simplify and solve the equation.
3) Simplify and solve the equation.
4) Simplify and solve the equation.
5) Simplify and solve the equation.
6) Simplify and solve the equation.
7) Simplify and solve the equation.
8) Simplify and solve the equation.
9) Simplify and solve the equation.
10) Standardized Assessment Practice
Choose the equivalent equation.
