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Solving Equations with Fraction Coefficients

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Solving Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\frac{1}{8}\phantom{\rule{0.1em}{0ex}}x+\frac{1}{2}=\frac{1}{4}.\)

 Solving Equations with Fraction Coefficients
To isolate the \(x\) term, subtract \(\frac{1}{2}\) from both sides.Solving Equations with Fraction Coefficients
Simplify the left side.Solving Equations with Fraction Coefficients
Change the constants to equivalent fractions with the LCD.Solving Equations with Fraction Coefficients
Subtract.Solving Equations with Fraction Coefficients
Multiply both sides by the reciprocal of \(\frac{1}{8}\).Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients

This method worked fine, but many students don’t 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 with no fractions. This process is called clearing the equation of fractions. Let’s solve the same equation again, but this time use the method that clears the fractions.

Example

Solve: \(\frac{1}{8}\phantom{\rule{0.1em}{0ex}}x+\frac{1}{2}=\frac{1}{4}.\)

Solution

Find the least common denominator of all the fractions in the equation.Solving Equations with Fraction Coefficients
Multiply both sides of the equation by that LCD, 8. This clears the fractions.Solving Equations with Fraction Coefficients
Use the Distributive Property.Solving Equations with Fraction Coefficients
Simplify — and notice, no more fractions!Solving Equations with Fraction Coefficients
Solve using the General Strategy for Solving Linear Equations.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Check: Let \(x=-2\)

 

Solving Equations with Fraction Coefficients

 

Notice in the example above that once we cleared the equation of fractions, the equation was like those we solved earlier in this tutorial. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

How to Solve equations with fraction coefficients by clearing the fractions.

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

Example

Solve: \(7=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x+\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x-\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x.\)

Solution

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

Find the least common denominator of all the fractions in the equation.Solving Equations with Fraction Coefficients
Multiply both sides of the equation by 12.Solving Equations with Fraction Coefficients
Distribute.Solving Equations with Fraction Coefficients
Simplify — and notice, no more fractions!Solving Equations with Fraction Coefficients
Combine like terms.Solving Equations with Fraction Coefficients
Divide by 7.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Check: Let \(x=12.\) 
Solving Equations with Fraction Coefficients 

In the next example, we’ll have variables and fractions on both sides of the equation.

Example

Solve: \(x+\frac{1}{3}=\frac{1}{6}\phantom{\rule{0.1em}{0ex}}x-\frac{1}{2}.\)

Solution

Find the LCD of all the fractions in the equation.Solving Equations with Fraction Coefficients
Multiply both sides by the LCD.Solving Equations with Fraction Coefficients
Distribute.Solving Equations with Fraction Coefficients
Simplify — no more fractions!Solving Equations with Fraction Coefficients
Subtract \(x\) from both sides.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Subtract 2 from both sides.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Divide by 5.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Check: Substitute \(x=-1.\) 
Solving Equations with Fraction Coefficients 

In the example below, we’ll start by using the Distributive Property. This step will clear the fractions right away!

Example

Solve: \(1=\frac{1}{2}\left(4x+2\right).\)

Solution

 Solving Equations with Fraction Coefficients
Distribute.Solving Equations with Fraction Coefficients
Simplify. Now there are no fractions to clear!Solving Equations with Fraction Coefficients
Subtract 1 from both sides.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Divide by 2.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Check: Let \(x=0.\) 
Solving Equations with Fraction Coefficients 

Many times, there will still be fractions, even after distributing.

Example

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

Solution

 Solving Equations with Fraction Coefficients
Distribute.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Multiply by the LCD, 4.Solving Equations with Fraction Coefficients
Distribute.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Collect the \(y\) terms to the left.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Collect the constants to the right.Solving Equations with Fraction Coefficients
Simplify.Solving Equations with Fraction Coefficients
Check: Substitute \(9\) for \(y.\) 
Solving Equations with Fraction Coefficients 

Optional Video: Solve an Equation with Fractions with Variable Terms on Both Sides

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