how to solve variable equations with fractions

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Course: Algebra 1 > Unit 2

Equation with variables on both sides: fractions

Equation with the variable in the denominator

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$how to solve variable equations with fractions$

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ACT Math : How to solve for a variable as part of a fraction

Study concepts, example questions & explanations for act math, all act math resources, example questions, example question #1 : how to solve for a variable as part of a fraction.

$how to solve variable equations with fractions$

Example Question #2 : How To Solve For A Variable As Part Of A Fraction

$how to solve variable equations with fractions$

Step 1: Set up the equation

$how to solve variable equations with fractions$

Step 2: Solve for D

$how to solve variable equations with fractions$

Example Question #3 : How To Solve For A Variable As Part Of A Fraction

$how to solve variable equations with fractions$

Solve for x:

$how to solve variable equations with fractions$

Example Question #7 : How To Solve For A Variable As Part Of A Fraction

$how to solve variable equations with fractions$

none of these

cross multiply:

(6)(19) = 9x

$\dpi{100} \small \frac{4 }{x} = \frac{2}{25}$

Cross multiply:

$\dpi{100} \small 4 \times 25 = 2x$

The numerator of a fraction is the sum of 4 and 5 times the denominator. If you divide the fraction by 2, the numerator is 3 times the denominator. Find the simplified version of the fraction.

$how to solve variable equations with fractions$

Let numerator = N and denominator = D.

According to the first statement,

N = (D x 5) + 4.

According to the second statement, N / 2 = 3 * D.

Let's multiply the second equation by –2 and add itthe first equation:

+[N = (D x 5) + 4]

–6D + (D x 5) + 4 = 0

–1D + 4 = 0

Thus, N = 24.

Therefore, N/D = 24/4 = 6.

Example Question #4 : How To Solve For A Variable As Part Of A Fraction

Solve the following equation for the given variable:

$how to solve variable equations with fractions$

To solve this equation we have to multiply both sides by the denominator to get rid of the fraction.

Doing this yields

$how to solve variable equations with fractions$

Then to solve the last step is to isolate the variable by dividing both sides by 12. Thus,

$how to solve variable equations with fractions$

Example Question #5 : How To Solve For A Variable As Part Of A Fraction

$how to solve variable equations with fractions$

When the equation is cross multiplied, it becomes

$how to solve variable equations with fractions$

Example Question #6 : How To Solve For A Variable As Part Of A Fraction

$how to solve variable equations with fractions$

Reduce any fractions in your final answer.

$how to solve variable equations with fractions$

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Equations with Fractions - Examples & Practice - Expii

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How To Solve Equations With Fractions

What is an equation.

An equation is a statement used in mathematics to show that two items are equal. In fractions, an equation is used to find the value of a fraction when one or more of its parts are unknown.

If you’re stuck on a math problem that involves fractions, don’t worry! In this article, we’ll show you how to solve equations with fractions step by step.

Step 1: Find the least common denominator

Firstly we need to find the least common denominator (LCD) of the fractions found in the equation, which is the smallest number that can be a common denominator for both of the fractions. For this equation the LCD is 12 as this is the lowest common multiple of 4 and 6.

Step 2: Multiply the least common denominator

Multiply the LCD to both sides of the equation.

Step 3: Simplify the equation

Simplify both sides of the equation and make sure we’re only working with whole numbers.

Step 4: Simplify until there's one term on both sides

Move all terms with the variable on one side and further simplify both sides of the Equations so we have one term on both sides.

Step 5: Divide the coefficient on both sides

Once the variable is isolated on one side, divide the coefficient on both sides to solve for the unknown variable.

Examples of how to solve equations with fractions

Q1) Find the value of x in:

$how to solve variable equations with fractions$

First, let’s find the least common denominator (LCD) of the fractions:

6=2×3 15=3×5 LCD:2×3×5=30

Multiply 30 on both sides of the equation. Make sure to simplify after distributing 30.

$how to solve variable equations with fractions$

Move 6x on on the left-hand side of the equation to isolate the term with the variable.

20x-6x=60 14x=60

Divide 14 on both sides of the equation to solve for x.

$how to solve variable equations with fractions$

Q2) An unknown fraction is added to 1 and we divide the sum by 3. The result is equal to 3/4. What is the value of the unknown fraction?

We can let n be the unknown number. We can set up the equation to solve the problem.

First, look for the least common denominator (LCD). Since 3 and 4 don’t share any common factors, we find their product.

Multiplying 12 to both sides of the equation, we have:

$how to solve variable equations with fractions$

Move 4 on the right-hand side of the equation.

4n=9-4 4n=5

Divide both sides by 4 to isolate n.

$how to solve variable equations with fractions$

Hence, the unknown fraction is equal to 5/4.

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$how to solve variable equations with fractions$

How to solve equations with fractions

$how to solve variable equations with fractions$

What is a fractional equation?

In this lesson we’ll look at how to solve equations with numerical fractions as coefficients and terms.

Remember that multiplying a fraction by its reciprocal will always give you a value of ???1???.

For example ???4/5??? has a reciprocal of ???5/4??? because

???\frac{4}{5}\cdot\frac{5}{4}=1???

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I create online courses to help you rock your math class. Read more.

To clear a fraction from an equation, multiply all of the terms on both sides of the equation by the fraction’s denominator.

For example, to clear the ???2??? from the fraction in ???5x+1/2=12???, multiply the equation by ???2??? on both sides.

???2\left(5x+\frac{1}{2}=12\right)???

???2(5x)+2\left(\frac{1}{2}\right)=2(12)???

???10x+1=24???

How to solve equations when there’s a fraction somewhere in the equation

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Take the course

Want to learn more about algebra 2 i have a step-by-step course for that. :), clearing the fraction from the equation in order to solve for the variable.

Solve for the variable.

???\frac{4}{5}n=20???

To get rid of a fractional coefficient, we have to multiply both sides by its reciprocal, because that’ll make the fraction ???1???.

???\frac{5}{4}\cdot\frac{4}{5}n=\frac{5}{4}\cdot20???

???\frac{20}{20}n=\frac{100}{4}???

???1n=25???

If you have a fractional coefficient and another term, you can isolate the term with the variable and then multiply both sides by the reciprocal of the fractional coefficient.

$how to solve variable equations with fractions$

???\frac{4}{7}x+14=22???

First isolate the fractional term.

???\frac{4}{7}x+14-14=22-14???

???\frac{4}{7}x=8???

Now get rid of the fractional coefficient by multiplying both sides of the equation by the reciprocal of ???4/7???.

???\frac{7}{4}\cdot\frac{4}{7}x=\frac{7}{4}\cdot8???

???\frac{28}{28}x=\frac{56}{4}???

???1x=14???

We could also do this same problem by first clearing the fraction. In order to get rid of the fraction, we have to multiply every term in the equation by its denominator.

???7\left(\frac{4}{7}x+14=22\right)???

???7\cdot\frac{4}{7}x+7\cdot14=7\cdot22???

???4x+98=154???

Now we can solve for the variable using inverse operations.

???4x+98-98=154-98???

???4x=56???

???\frac{4x}{4}=\frac{56}{4}???

$how to solve variable equations with fractions$

Get access to the complete Algebra 2 course

Solving Equations with Fractions

I know fractions are difficult, but with these easy step-by step instructions you'll be solving equations with fractions in no time.

Do you start to get nervous when you see fractions? Do you have to stop and review all the rules for adding, subtracting, multiplying and dividing fractions?

If so, you are just like almost every other math student out there! But... I am going to make your life so much easier when it comes to solving equations with fractions!

Our first step when solving these equations is to get rid of the fractions because they are not easy to work with!

Let see what happens with a typical two-step equation with the distributive property.

In this problem, we would typically distribute the 3/4 throughout the parenthesis and then solve. Let's see what happens:

Yuck! That just made this problem worse! Now we have two fractions to contend with and that means subtracting fractions and multiplying fractions.

So... let's stop here and say,

We DO NOT want to do this! DO NOT distribute fractions.

We are going to learn how to get rid of the fractions and make this much more simple!

So... what do we do? We are going to get rid of just the denominator in the fraction, so we will be left with the numerator, or just an integer!

I know, easier said than done! It's really not hard, but before I get into it, I want to go over one algebra definition.

We need to discuss the word term.

In Algebra, each term within an equation is separated by a plus (+) sign, minus (-) sign or an equals sign (=). Variable or quantities that are multiplied or divided are considered the same term.

That last example is the most important to remember. If a quantity is in parentheses, it it considered one term!

Let's look at a few examples of how to solve these crazy looking problems!

Example 1 - Equations with Fractions

$Solving equations with fractions by eliminating the fractions$

Take a look at this example on video if you are feeling overwhelmed.

Hopefully you were able to follow that example. I know it's tough, but if you can get rid of the fraction, it will make these problems so much easier. Keep going, you'll get the hang of it!

In the next example, you will see two fractions. Since they have the same denominator, we will multiply by the denominator and get rid of both fractions.

Example 2 - Equations with Fractions with the Same Denominator

$how to solve variable equations with fractions$

Did you notice how multiplying by 2 (the denominator of both fractions) allowed us to get rid of the fractions? This is the best way to deal with equations that contain fractions.

In the next example, you will see what happens when you have 2 fractions that have different denominators.

We still want to get rid of the fractions all in one step. Therefore, we need to multiply all terms by the least common multiple. Remember how to find the LCM? If not, check out the LCM lesson here .

Example 3 - Equations with Two Fractions with Different Denominators

$Solving equations with two fractions that have different denominators$

Yes, the equations are getting harder, but if you take it step-by-step, you will arrive at the correct solution. Keep at it - I know you'll get it!

$how to solve variable equations with fractions$

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Module 8: Growth Models

Solving equations by clearing fractions, learning outcomes.

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.

https://ohm.lumenlearning.com/multiembedq.php?id=71948&theme=oea&iframe_resize_id=mom1

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

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.

https://ohm.lumenlearning.com/multiembedq.php?id=71948&theme=oea&iframe_resize_id=mom2

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.

https://ohm.lumenlearning.com/multiembedq.php?id=142514&theme=oea&iframe_resize_id=mom3

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.

https://ohm.lumenlearning.com/multiembedq.php?id=142542&theme=oea&iframe_resize_id=mom25

$how to solve variable equations with fractions$

IMAGES

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Algebra
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VIDEO

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COMMENTS

Equation with variables on both sides: fractions
The general rule for solving equations with fractions — whether it be only on one side or both — is to try to get rid of all of them. The most common way to
ACT Math : How to solve for a variable as part of a fraction
To solve an equation with a variable in a fraciton, treat the denominator as a constant value and multiply both sides of the equation by the denominator in
Solving an Equation in One Variable with Fractions (Keep Fractions 1
This video provides two examples of how to solve a linear equation in one variable that contains fractions. The fractions are not cleared
How To Solve Linear Equations With Fractions
This algebra video tutorial explains how to solve linear equations with fractions.
Equations with Fractions
Isolate x on one side of the equation (whether x is part of the fraction or not). · If the only terms in your equation are fractions, you can cross-multiply them
Easy Guide on How to Solve Equations With Fractions
Step 1: Find the least common denominator · Step 2: Multiply the least common denominator · Step 3: Simplify the equation · Step 4: Simplify until
How to solve equations with fractions
In this lesson we'll look at how to solve equations with numerical fractions as coefficients and terms. To clear a fraction from an equation
Solving Equations With Fractions
In Algebra, each term within an equation is separated by a plus (+) sign, minus (-) sign or an equals sign (=). Variable or quantities that are multiplied or
4.9: Solving Equations with Fractions
Applications · 1. Set up a Variable Dictionary. Our variable dictionary will take the form of a well labeled diagram. · 2. Set up an Equation. The
Solving Equations By Clearing Fractions
Find the least common denominator of all the fractions in the equation. · Multiply both sides of the equation by that LCD. · Isolate the variable terms on one