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Inverse operations

Solve equations with fractions

Here you will learn about how to solve equations with fractions, including solving equations with one or more operations. You will also learn about solving equations with fractions where the unknown is the denominator of a fraction.

Students will first learn how to solve equations with fractions in 7th grade as part of their work with expressions and equations and expand that knowledge in 8th grade.

What are equations with fractions?

Equations with fractions involve solving equations where the unknown variable is part of the numerator and/or denominator of a fraction.

The numerator (top number) in a fraction is divided by the denominator (bottom number).

To solve equations with fractions, you will use the “balancing method” to apply the inverse operation to both sides of the equation in order to work out the value of the unknown variable.

The inverse operation of addition is subtraction.

The inverse operation of subtraction is addition.

The inverse operation of multiplication is division.

The inverse operation of division is multiplication.

For example,

\begin{aligned} \cfrac{2x+3}{5} \, &= 7\\ \colorbox{#cec8ef}{$\times \, 5$} \; & \;\; \colorbox{#cec8ef}{$\times \, 5$} \\\\ 2x+3&=35 \\ \colorbox{#cec8ef}{$-\,3$} \; & \;\; \colorbox{#cec8ef}{$- \, 3$} \\\\ 2x & = 32 \\ \colorbox{#cec8ef}{$\div \, 2$} & \; \; \; \colorbox{#cec8ef}{$\div \, 2$}\\\\ x & = 16 \end{aligned}

What are equations with fractions?

Common Core State Standards

How does this relate to 7th grade and 8th grade math?

  • Grade 7: Expressions and Equations (7.EE.A.1) Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
  • Grade 8: Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • Grade 8: Expressions and Equations (8.EE.C.7b) Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

How to solve equations with fractions

In order to solve equations with fractions:

Identify the operations that are being applied to the unknown variable.

Apply the inverse operations, one at a time, to both sides of the equation.

Write the final answer, checking that it is correct.

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

Solve equations with fractions examples

Example 1: equations with one operation.

Solve for x \text{: } \cfrac{x}{5}=4 .

The unknown is x.

Looking at the left hand side of the equation, the x is divided by 5.

\cfrac{x}{5}

2 Apply the inverse operations, one at a time, to both sides of the equation.

The inverse of “dividing by 5 ” is “multiplying by 5 ”.

You will multiply both sides of the equation by 5.

Solve equations with fractions example 1

3 Write the final answer, checking that it is correct.

The final answer is x=20.

You can check the answer by substituting the answer back into the original equation.

\cfrac{20}{5}=20\div5=4

Example 2: equations with one operation

Solve for x \text{: } \cfrac{x}{3}=8 .

Looking at the left hand side of the equation, the x is divided by 3.

\cfrac{x}{3}

The inverse of “dividing by 3 ” is “multiplying by 3 ”.

You will multiply both sides of the equation by 3.

Solve equations with fractions example 2

The final answer is x=24.

\cfrac{24}{3}=24\div3=8

Example 3: equations with two operations

Solve for x \text{: } \cfrac{x \, + \, 1}{2}=7 .

Looking at the left hand side of the equation, 1 is added to x and then divided by 2 (the denominator of the fraction).

\cfrac{x \, + \, 1}{2}

First, clear the fraction by multiplying both sides of the equation by 2.

Then, subtract 1 from both sides.

Solve equations with fractions example 3

The final answer is x=13.

\cfrac{13 \, +1 \, }{2}=\cfrac{14}{2}=14\div2=7

Example 4: equations with two operations

Solve for x \text{: } \cfrac{x}{4}-2=3 .

Looking at the left hand side of the equation, x is divided by 4 and then 2 is subtracted.

\cfrac{x}{4}-2

First, add 2 to both sides of the equation.

Then, multiply both sides of the equation by 4.

Solve equations with fractions example 4

\cfrac{20}{4}-2=20\div4-2=5-2=3

Example 5: equations with three operations

Solve for x \text{: } \cfrac{3x}{5}+1=7 .

Looking at the left hand side of the equation, x is multiplied by 3, then divided by 5 , and then 1 is added.

\cfrac{3x}{5}+1

First, subtract 1 from both sides of the equation.

Then, multiply both sides of the equation by 5.

Finally, divide both sides by 3.

Solve equations with fractions example 5

The final answer is x=10.

\cfrac{3 \, \times \, 10}{5}+1=\cfrac{30}{5}+1=6+1=7

Example 6: equations with three operations

Solve for x \text{: } \cfrac{2x-1}{7}=3 .

Looking at the left hand side of the equation, x is multiplied by 2, then 1 is subtracted, and the last operation is divided by 7 (the denominator).

\cfrac{2x-1}{7}

First, multiply both sides of the equation by 7.

Next, add 1 to both sides.

Solve equations with fractions example 6

The final answer is x=11.

\cfrac{2 \, \times \, 11-1}{7}=\cfrac{22-1}{7}=\cfrac{21}{7}=3

Example 7: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{24}{x}=6 .

Looking at the left hand side of the equation, x is the denominator. 24 is divided by x.

\cfrac{24}{x}

You need to multiply both sides of the equation by x.

Then, you can divide both sides by 6.

Solve equations with fractions example 7

The final answer is x=4.

\cfrac{24}{4}=24\div4=6

Example 8: equations with the unknown as the denominator

Solve for x \text{: } \cfrac{18}{x}-6=3 .

Looking at the left hand side of the equation, x is the denominator. 18 is divided by x , and then 6 is subtracted.

\cfrac{18}{x}-6

First, add 6 to both sides of the equation.

Then, multiply both sides of the equation by x.

Finally, divide both sides by 9.

Solve equations with fractions example 8

The final answer is x=2.

\cfrac{18}{2}-6=9-6=3

Teaching tips for solving equations with fractions

  • When students first start working through practice problems and word problems, provide step-by-step instructions to assist them with solving linear equations.
  • Introduce solving equations with fractions with one-step problems, then two-step problems, before introducing multi-step problems.
  • Students will need lots of practice with solving linear equations. These standards provide the foundation for work with future linear equations in Algebra I and II.
  • Provide opportunities for students to explain their thinking through writing. Ensure that they are using key vocabulary, such as, absolute value, coefficient, equation, common factors, inequalities, simplify, etc.

Easy mistakes to make

  • The solution to an equation can be any type of number The unknowns do not have to be integers (whole numbers and their negative opposites). The solutions can be fractions or decimals. They can also be positive or negative numbers.
  • The unknown of an equation can be on either side of the equation The unknown, represented by a letter, is often on the left hand side of the equations; however, it doesn’t have to be. It could also be on the right hand side of an equation.

Solve equations with fractions image 2

  • Lowest common denominator (LCD) It is common to get confused between solving equations involving fractions and adding and subtracting fractions. When adding and subtracting, you need to work out the lowest/least common denominator (sometimes called the least common multiple or LCM). When you solve equations involving fractions, multiply both sides of the equation by the denominator of the fraction.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Substitution
  • Linear equations
  • Writing linear equations
  • Solving equations
  • Identity math
  • One step equations

Practice solve equations with fractions questions

1. Solve: \cfrac{x}{6}=3

GCSE Quiz False

You will multiply both sides of the equation by 6, because the inverse of “dividing by 6 ” is “multiplying by 6 ”.

Solve equations with fractions practice question 1

The final answer is x = 18.

\cfrac{18}{6}=18 \div 6=3

2. Solve: \cfrac{x \, + \, 4}{2}=7

Then subtract 4 from both sides.

Solve equations with fractions practice question 2

The final answer is x = 10.

\cfrac{10 \, + \, 4}{2}=\cfrac{14}{2}=14 \div 2=7

3. Solve: \cfrac{x}{8}-5=1

First, add 5 to both sides of the equation.

Then multiply both sides of the equation by 8.

Solve equations with fractions practice question 3

The final answer is x = 48.

\cfrac{48}{8}-5=48 \div 8-5=1

4. Solve: \cfrac{3x \, + \, 2}{4}=2

First, multiply both sides of the equation by 4.

Next, subtract 2 from both sides.

Solve equations with fractions practice question 4

The final answer is x = 2.

\cfrac{3 \, \times \, 2+2}{4}=\cfrac{6 \, + \, 2}{4}=\cfrac{8}{4}=8 \div 4=2

5. Solve: \cfrac{4x}{7}-2=6

Then multiply both sides of the equation by 7.

Finally, divide both sides by 4.

Solve equations with fractions practice question 5

The final answer is x = 14.

\cfrac{4 \, \times \, 14}{7}-2=\cfrac{56}{7}-2=56 \div 7-2=6

6. Solve: \cfrac{42}{x}=7

Then you divide both sides by 7.

Solve equations with fractions practice question 6

The final answer is x = 6.

\cfrac{42}{6}=42 \div 6=7

Solve equations with fractions FAQs

Yes, you still follow the order of operations when solving equations with fractions. You will start with any operations in the numerator and follow PEMDAS (parenthesis, exponents, multiply/divide, add/subtract), followed by any operations in the denominator. Then you will solve the rest of the equation as usual.

The next lessons are

  • Inequalities
  • Types of graphs
  • Coordinate plane
  • Number patterns
  • Algebraic expressions
  • Fractions operations

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Module 5: Multi-Step Linear Equations

Solving equations by clearing fractions, learning outcomes.

  • Use the least common denominator to eliminate fractions from a linear equation before solving it
  • Solve equations with fractions that require several steps

You may feel overwhelmed when you see fractions in an equation, so we are going to show a method to solve equations with fractions where you use the common denominator to eliminate the fractions from an equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions.

Pay attention to the fact that each term in the equation gets multiplied by the least common denominator. That’s what makes it equal to the original!

Solve: [latex]\Large\frac{1}{8}\normalsize x+\Large\frac{1}{2}=\Large\frac{1}{4}[/latex]

In the last example, the least common denominator was [latex]8[/latex]. Now it’s your turn to find an LCD, and clear the fractions before you solve these linear equations.

Notice that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve!

Solve equations by clearing the Denominators

  • Find the least common denominator of all the fractions in the equation.
  • Multiply both sides of the equation by that LCD. This clears the fractions.
  • Isolate the variable terms on one side, and the constant terms on the other side.
  • Simplify both sides.
  • Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[/latex].

Here’s an example where you have three variable terms. After you clear fractions with the LCD, you will simplify the three variable terms, then isolate the variable.

Solve: [latex]7=\Large\frac{1}{2}\normalsize x+\Large\frac{3}{4}\normalsize x-\Large\frac{2}{3}\normalsize x[/latex]

Show Solution

Solution: We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Now here’s a similar problem for you to try. Clear the fractions, simplify, then solve.

One of the most common mistakes when you clear fractions is forgetting to multiply BOTH sides of the equation by the LCD. If your answer doesn’t check, make sure you have multiplied both sides of the equation by the LCD.

In the next example, we’ll have variables and fractions on both sides of the equation. After you clear the fractions using the LCD, you will see that this equation is similar to ones with variables on both sides that we solved previously. Remember to choose a variable side and a constant side to help you organize your work.

Solve: [latex]x+\Large\frac{1}{3}=\Large\frac{1}{6}\normalsize x-\Large\frac{1}{2}[/latex]

Now you can try solving an equation with fractions that has variables on both sides of the equal sign. The answer may be a fraction.

In the following video we show another example of how to solve an equation that contains fractions and variables on both sides of the equal sign.

In the next example, we start with an equation where the variable term is locked up in some parentheses and multiplied by a fraction. You can clear the fraction, or if you use the distributive property it will eliminate the fraction.  Can you see why?

Solve: [latex]1=\Large\frac{1}{2}\normalsize\left(4x+2\right)[/latex]

Now you can try solving an equation that has the variable term in parentheses that are multiplied by a fraction.

  • Question ID 142514, 142542. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
  • Solve a Linear Equation with Parentheses and a Fraction 2/3(9x-12)=8+2x. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/1dmEoG7DkN4 . License : CC BY: Attribution
  • Ex 1: Solve an Equation with Fractions with Variable Terms on Both Sides. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/G5R9jySFMpw . License : CC BY: Attribution
  • Question ID 71948. Authored by : Alyson Day. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
  • Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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how to solve equation with fractions on both sides

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  • PowerPoints

Fractions on Both Sides Equations

In this lesson we look at How To Solve Equations With Fractions on Both Sides.

Prior to doing this lesson, you need to know how to solve equations with letters and/or brackets on both sides.

If you need to find out how to do Variable Letters On Both Sides Equations, then go through our previous lesson on this at the following link:

Solving Variable Letter on Both Sides Lesson

The method for “Fractions Both Sides” is basically the same as solving equations with the letter on both sides, except that we need to do an extra cross multiplying step at the beginning.

Prior to going through our lesson on “Fractions Both Sides” :

We also recommend you check that you understand the material from the following previous lessons:

Cross Multiplying Ratio Equations Lesson

Expanding Brackets Using the Distributive Rule Lesson

   

Fractions Both Sides Equation

Here is a typical “Fractions Both Sides” Equation.

These types of Equations are done the same way as “Variable Letter on Both Sides” equations, but we have one extra beginning step at the start.

Fractions Both Sides Working Out Steps

The cross multiplying is the same as we do for Ratios.

Eg. The Ratio Equation 3/4 = 6/8 cross multiplied gives

8×3 = 6×4 which is a true equation,

and so cross multiplying is a legitimate method.

Fractions Both Sides – Example One

Here is how we do the Cross Multipying Step for a typical “Fractions Both Sides” Equation:

Note that we need to use Brackets to make sure ALLL the terms in the top line “numerator” get multiplied.

After we have completed the Cross Multiplying, we are left with a normal “Variable Letter Both Sides” equation to solve.

We solve this as follows:

If you do not know how to Solve for Variable Letter on Both Sides, then see our previous lesson on this at the following link:

Fractions Both Sides – Example Two

In this example, only one side has a fraction.

We do a little “trick” which involves putting the non-fraction side as being a fraction that is / 1

We can then do the Cross Multiplying Step.

Here is the example, showing the Cross Multiplying Step

Here are the remaining steps that are required to get to the final answer:

Videos About Fractions Both Sides Equations

These two videos show how to use the Proportions Cross Multiplying Method to Solve Equations.

And here is the second video you need to watch.

Related Items

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how to solve equation with fractions on both sides

How Do You Solve an Equation with Variables on Both Sides and Fractions?

Trying to solve an equation with variables and fractions on both sides of the equation? You can bet it involves finding a common denominator! To see what it takes, watch this tutorial.

  • linear equation
  • variable on both sides
  • distributive property
  • multiplication property of equality
  • solve by subtraction
  • solve by division

Background Tutorials

Introduction to algebraic expressions.

What is a Variable?

What is a Variable?

You can't do algebra without working with variables, but variables can be confusing. If you've ever wondered what variables are, then this tutorial is for you!

Adding Real Numbers

How Do You Find a Common Denominator and a Least Common Denominator?

How Do You Find a Common Denominator and a Least Common Denominator?

This tutorial gives you some practice finding a common denominator and the least common denominator of three fractions. There's only one least common denominator, but there are many common denominators. This tutorial gives you one. Can you find another?

Subtracting Real Numbers

How Do You Subtract a Whole Number from a Fraction?

How Do You Subtract a Whole Number from a Fraction?

Subtracting a whole number from a fraction can be tricky. Luckily, watching this tutorial can make this subtraction no big deal!

The Distributive Property

How Do You Use the Distributive Property to Simplify an Expression?

How Do You Use the Distributive Property to Simplify an Expression?

In this tutorial you'll see how to apply the distributive property. Remember that this is important when you are trying to simplify an expression and get rid of parentheses!

Properties of Equality

What's the Multiplication Property of Equality?

What's the Multiplication Property of Equality?

Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the multiplication property of equality, and it lets you multiply both sides of an equation by the same number. Watch the video to see it in action!

Solving Equations with Variables on Both Sides

How Do You Solve an Equation with Variables on Both Sides?

How Do You Solve an Equation with Variables on Both Sides?

Trying to solve an equation with variables on both sides of the equal sign? Figure out how to get those variables together and find the answer with this tutorial!

Further Exploration

Solving two-step equations.

How Do You Solve a Multi-Step Equation with Fractions by Multiplying Away the Fraction?

How Do You Solve a Multi-Step Equation with Fractions by Multiplying Away the Fraction?

Trying to solve an equation involving a fraction? Just multiply the fraction away and then perform the order of operations in reverse! See how in this tutorial.

How Do You Solve a Multi-Step Equation with Fractions Using Reverse Order of Operations and Reciprocals?

How Do You Solve a Multi-Step Equation with Fractions Using Reverse Order of Operations and Reciprocals?

Trying to solve an equation involving a fraction? Just perform the order of operations in reverse! See how in this tutorial.

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How to Solve Equations with Variables on Both Sides

Last Updated: March 11, 2023 Fact Checked

This article was co-authored by JohnK Wright V . JohnK Wright V is a Certified Math Teacher at Bridge Builder Academy in Plano, Texas. With over 20 years of teaching experience, he is a Texas SBEC Certified 8-12 Mathematics Teacher. He has taught in six different schools and has taught pre-algebra, algebra 1, geometry, algebra 2, pre-calculus, statistics, math reasoning, and math models with applications. He was a Mathematics Major at Southeastern Louisiana and he has a Bachelor of Science from The University of the State of New York (now Excelsior University) and a Master of Science in Computer Information Systems from Boston University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 185,941 times.

To study algebra, you will see equations that have a variable on one side, but later on you will often see equations that have variables on both sides. The most important thing to remember when solving such equations is that whatever you do to one side of the equation, you must do to the other side. Using this rule, it is easy to move variables around so that you can isolate them and use basic operations to find their value.

Solving Equations with One Variable on Both Sides

Step 1 Apply the distributive property, if necessary.

Solving System Equations with Two Variables

Step 1 Isolate a variable in one equation.

Solving Example Problems

Step 1 Try this problem using the distributive property with one variable:

Community Q&A

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  • ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-distributive-property/v/the-distributive-property
  • ↑ https://www.virtualnerd.com/algebra-1/linear-equations-solve/variables-both-sides-equations/variables-both-sides-solution/variables-grouping-symbols-both-sides
  • ↑ https://www.youtube.com/watch?v=hrAOSknrYiI&t=296s
  • ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-evaluating-expressions/v/expression-terms-factors-and-coefficients
  • ↑ https://www.virtualnerd.com/pre-algebra/linear-functions-graphing/system-of-equations/solving-systems-equations/two-equations-two-variables-substitution

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Mathematics LibreTexts

3.6: Solving Equations with Fractions or Decimals

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Learning Objectives

By the end of this section, you will be able to:

  • Solve equations with fraction coefficients
  • Solve equations with decimal coefficients

Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, \(\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}\).

This method worked fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

Exercise \(\PageIndex{1}\): How to Solve Equations with Fraction Coefficients

Solve: \(\frac{1}{6}y - \frac{1}{3} = \frac{5}{6}\)

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Find the least common denominator of all the fractions in the equation.” The text in the second cell reads: “What is the LCD of 1/6, 1/3, and 5/6?” The third cell contains the equation one-sixth y minus 1/3 equals 5/6, with LCD equals 6 written next to it.

Exercise \(\PageIndex{2}\)

Solve: \(\frac{1}{4}x + \frac{1}{2} = \frac{5}{8}\)

\(x= \frac{1}{2}\)

Exercise \(\PageIndex{3}\)

Solve: \(\frac{1}{8}x + \frac{1}{2} = \frac{1}{4}\)

Notice in Exercise \(\PageIndex{1}\), once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

STRATEGY TO SOLVE EQUATIONS WITH FRACTION COEFFICIENTS.

  • Find the least common denominator of all the fractions in the equation.
  • Multiply both sides of the equation by that LCD. This clears the fractions.
  • Solve using the General Strategy for Solving Linear Equations.

Exercise \(\PageIndex{4}\)

Solve: \(6 = \frac{1}{2}v + \frac{2}{5}v - \frac{3}{4}v\)

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Exercise \(\PageIndex{5}\)

Solve: \(7 = \frac{1}{2}x + \frac{3}{4}x - \frac{2}{3}x\)

Exercise \(\PageIndex{6}\)

Solve: \(-1 = \frac{1}{2}u + \frac{1}{4}u - \frac{2}{3}u\)

\(u = -12\)

In the next example, we again have variables on both sides of the equation.

Exercise \(\PageIndex{7}\)

Solve: \(a + \frac{3}{4} = \frac{3}{8}a - \frac{1}{2}\)

Exercise \(\PageIndex{8}\)

Solve: \(x + \frac{1}{3} = \frac{1}{6}x - \frac{1}{2}\)

Exercise \(\PageIndex{9}\)

Solve: \(c + \frac{3}{4} = \frac{1}{2}c - \frac{1}{4}\)

In the next example, we start by using the Distributive Property. This step clears the fractions right away.

Exercise \(\PageIndex{10}\)

Solve: \(-5 = \frac{1}{4}(8x + 4)\)

Exercise \(\PageIndex{11}\)

Solve: \(-11 = \frac{1}{2}(6p + 2)\)

Exercise \(\PageIndex{12}\)

Solve: \(8 = \frac{1}{3}(9q + 6)\)

In the next example, even after distributing, we still have fractions to clear.

Exercise \(\PageIndex{13}\)

Solve: \(\frac{1}{2}(y - 5) = \frac{1}{4}(y - 1)\)

Exercise \(\PageIndex{14}\)

Solve: \(\frac{1}{5}(n + 3) = \frac{1}{4}(n + 2)\)

Exercise \(\PageIndex{15}\)

Solve: \(\frac{1}{2}(m - 3) = \frac{1}{4}(m - 7)\)

Exercise \(\PageIndex{16}\)

Solve: \(\frac{5x - 3}{4} = \frac{x}{2}\)

Exercise \(\PageIndex{17}\)

Solve: \(\frac{4y - 7}{3} = \frac{y}{6}\)

Exercise \(\PageIndex{18}\)

Solve: \(\frac{-2z - 5}{4} = \frac{z}{8}\)

Exercise \(\PageIndex{19}\)

Solve: \(\frac{a}{6} + 2 = \frac{a}{4} + 3\)

Exercise \(\PageIndex{20}\)

Solve: \(\frac{b}{10} + 2 = \frac{b}{4} + 5\)

\(b = -20\)

Exercise \(\PageIndex{21}\)

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

Exercise \(\PageIndex{22}\)

Solve: \(\frac{4q + 3}{2}+ 6 = \frac{3q + 5}{4}\)

Exercise \(\PageIndex{23}\)

Solve: \(\frac{3r + 5}{6}+ 1 = \frac{4r + 3}{3}\)

Exercise \(\PageIndex{24}\)

Solve: \(\frac{2s + 3}{2}+ 1 = \frac{3s + 2}{4}\)

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, \(0.3 = \frac{3}{10}\) and \(0.17 = \frac{17}{100}\). So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Exercise \(\PageIndex{25}\)

Solve: \(0.06x + 0.02 = 0.25x - 1.5\)

Look at the decimals and think of the equivalent fractions.

\(0.06 = \frac { 6 } { 100 } \quad 0.02 = \frac { 2 } { 100 } \quad 0.25 = \frac { 25 } { 100 } \quad 1.5 = 1 \frac { 5 } { 10 }\)

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

Exercise \(\PageIndex{26}\)

Solve: \(0.14h + 0.12 = 0.35h - 2.4\)

Exercise \(\PageIndex{27}\)

Solve: \(0.65k - 0.1 = 0.4k - 0.35\)

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

Exercise \(\PageIndex{28}\)

Solve: \(0.25x + 0.05(x + 3) = 2.85\)

Exercise \(\PageIndex{29}\)

Solve: \(0.25n + 0.05(n + 5) = 2.95\)

Exercise \(\PageIndex{30}\)

Solve: \(0.10d + 0.05(d -5) = 2.15\)

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