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Simplifying fractions

Adding and subtracting fractions

This topic is relevant for:

## Algebraic Fractions

## What are algebraic fractions?

Algebraic fractions are fractions that contain at least one variable.

The following algebraic expressions are examples of algebraic fractions:

One step equation: \quad \quad \frac{x}{12}=4 A separate constant term: \quad \frac{x+1}{2x}+4=x

A quadratic equation: \quad\frac{3}{x+1}=x+5 A linear equation: \quad \frac{3x+4}{2x-5}=6

Step-by-step guide: Simplifying algebraic fractions

## How to solve equations including algebraic fractions

We need to be able to solve equations including algebraic fractions.

Let’s look at a simple example when \frac{8}{x}=2 .

We can substitute this into the original equation to prove that the answer is correct.

Here, \frac{8}{4}=2 so we have the correct answer.

We shall now consider more complicated cases when equations involve algebraic fractions.

In order to solve equations including algebraic fractions.

Convert each fraction so they all have a common denominator.

- Multiply the equation throughout by the common denominator.
- Solve the equation (linear or quadratic).

## Explain how to solve equations including algebraic fractions

## Algebraic fractions worksheet

## Algebraic fractions examples

Example 1: equation with one fraction.

Here, we only have one fraction and so we do not need to convert any other term into a fraction.

2 Multiply the equation throughout by the common denominator .

Multiplying the equation throughout by 3 (the denominator of the fractional term), we get

Make sure that you multiply every term in the equation by 3 .

3 Solve the equation (linear or quadratic) .

## Example 2: Equation with two fractions

Remember to use brackets to ensure that you multiply the entire numerator by 5 .

Remember to use brackets to ensure that you multiply the entire numerator by 2 .

Multiply the equation throughout by the common denominator .

Multiplying the equation throughout by 10 (the denominator of the fractional terms), we get

Solve the equation (linear or quadratic) .

## Example 3: Equation with x in the denominator

Here, we have one fraction so we do not need to find a common denominator.

Multiplying the equation throughout by x + 1 (the denominator of the fractional terms), we get

## Example 4: Equation with three fractions

We now have an equation which we can immediately simplify.

Multiplying the equation throughout by 6x (the denominator of the fractional terms), we get

## Example 5: Denominators are expressions in terms of x

Here, we need to find a common denominator for (x + 2) and (x − 4) .

The easiest way to do this is to multiply the two expressions together.

By multiplying each fraction by the denominator of the other fraction, we get

Multiplying the equation throughout by (x+2)(x-4) (the denominator of the fractional term), we get

## Example 6: Equation including a quadratic

Here, we have a single fraction and so we do not need to find a common denominator.

Multiplying the equation throughout by x (the denominator of the fractional terms), we get

Here, we have two possible solutions for x so we can check both:

## Common misconceptions

## Practice algebraic fractions questions

## Algebraic fractions GCSE questions

2. Use the quadratic formula to solve the equation

2x(x+2)+5x(x+3)=4(x+3)(x+2) or equivalent

3. Two shapes given below have the same area. Calculate the value for x .

Area of Triangle = \frac{2(x+8)}{2}=x+8

Area of Square = \left(\frac{3}{\sqrt{x}}\right)^{2}=\frac{9}{x}

## Learning checklist

- use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)
- simplify and manipulate algebraic fractions

## The next lessons are

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- Operations with Algebraic Fractions
- Preliminaries
- Quiz: Preliminaries
- Properties of Basic Mathematical Operations
- Quiz: Properties of Basic Mathematical Operations
- Multiplying and Dividing Using Zero
- Quiz: Multiplying and Dividing Using Zero
- Powers and Exponents
- Quiz: Powers and Exponents
- Square Roots and Cube Roots
- Quiz: Square Roots and Cube Roots
- Grouping Symbols
- Quiz: Grouping Symbols
- Divisibility Rules
- Quiz: Divisibility Rules
- Signed Numbers (Positive Numbers and Negative Numbers)
- Quiz: Signed Numbers (Positive Numbers and Negative Numbers)
- Quiz: Fractions
- Simplifying Fractions and Complex Fractions
- Quiz: Simplifying Fractions and Complex Fractions
- Quiz: Decimals
- Quiz: Percent
- Scientific Notation
- Quiz: Scientific Notation
- Quiz: Set Theory
- Variables and Algebraic Expressions
- Quiz: Variables and Algebraic Expressions
- Evaluating Expressions
- Quiz: Evaluating Expressions
- Quiz: Equations
- Ratios and Proportions
- Quiz: Ratios and Proportions
- Solving Systems of Equations (Simultaneous Equations)
- Quiz: Solving Systems of Equations (Simultaneous Equations)
- Quiz: Monomials
- Polynomials
- Quiz: Polynomials
- Quiz: Factoring
- What Are Algebraic Fractions?
- Quiz: Operations with Algebraic Fractions
- Inequalities
- Quiz: Inequalities
- Graphing on a Number Line
- Quiz: Graphing on a Number Line
- Absolute Value
- Quiz: Absolute Value
- Solving Equations Containing Absolute Value
- Coordinate Graphs
- Quiz: Coordinate Graphs
- Linear Inequalities and Half-Planes
- Quiz: Linear Inequalities and Half-Planes
- Quiz: Functions
- Quiz: Variations
- Introduction to Roots and Radicals
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- Quiz: Simplifying Square Roots
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- Quiz: Operations with Square Roots
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- Quiz: Solving Quadratic Equations
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Warning: Do not reduce through an addition or subtraction sign as shown here.

Multiplying algebraic fractions

Adding or subtracting algebraic fractions

Perform the indicated operation.

If there is a common variable factor with more than one exponent, use its greatest exponent.

Previous What Are Algebraic Fractions?

Next Quiz: Operations with Algebraic Fractions

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## Solving Equations with Algebraic Fractions

## Solving Equations with Algebraic Fractions

\(\frac{5}{x}\), \(\frac{2x-9}{7}\), and \(\frac{3x}{2}\)

\(\frac{x}{8}=7\)

\(8\cdot \frac{x}{8}=7\cdot 8\)

Not too hard, right? Let’s try one that’s a little bit more complicated.

\(\frac{3x}{4}-\frac{2x}{12}=7\)

\(\frac{3x\cdot 3}{4\cdot 3}=\frac{9x}{12}\)

\(\frac{9x}{12}-\frac{2x}{12}=7\)

Since our denominators are the same, we can now subtract our numerators, so \(9x-2x=7x\).

\(\frac{7x}{12}=7\)

Another way we can write this is:

\(\frac{7}{12}\cdot x=7\)

\(\frac{12}{7}\cdot \frac{7}{12}\cdot x=7\cdot \frac{12}{7}\)

So \(x=12\), which is our answer.

Let’s take a look at another problem.

\(\frac{4x}{5}=\frac{3x}{10}+9\)

\(10(\frac{4x}{5}=\frac{3x}{10}+9)\)

\(8x=3x+90\)

Now we can solve it like a normal algebra problem . So I’m going to subtract \(3x\) from both sides.

\(8x-3x=3x-3x+90\)

And then we divide both sides by 5.

\(\frac{5x}{5}=\frac{90}{5}\)

Before we go, I want to show you one more problem you might come across.

\(\frac{x+17}{5}=21\)

\(5\cdot \frac{x+17}{5}=21\cdot 5\)

\(x+17=105\)

So now we’re going to solve it like normal. We’ll subtract 17 from both sides,

\(x+17-17=105-17\) \(x=88\)

## Practice Questions

Solve the equation: \(\frac{2x}{3}-\frac{4x}{9}=6\).

\(\frac{3}{3}\cdot\frac{2x}{3}-\frac{4x}{9}=6\) \(\frac{6x}{9}-\frac{4x}{9}=6\) \(\frac{2x}{9}=6\)

Now, multiply both sides by 9.

\(9\cdot\frac{2x}{9}=6\cdot9\) \(2x=54\)

Finally, divide both sides by 2.

\(\frac{2x}{2}=\frac{54}{2}\) \(x=27\)

Solve the equation: \(\frac{5x}{6}=\frac{11x}{15}+2\).

To get the variable terms on the left side, subtract 22x from both sides of the equation.

\(25x-22x=22x+60-22x\) \(3x=60\)

\(\frac{3x}{3}=\frac{60}{3}\) \(x=20\)

Solve the equation: \(2x+79=5\).

\(9\cdot \frac{2x+7}{9}=5\cdot9\) \(2x+7=45\)

Now, subtract 7 from both sides.

\(\frac{2x}{2}=\frac{38}{2}\) \(x=19\)

The quotient of thirteen less than a number and 7 equals twice the number. What is the number?

\(7\cdot\frac{x-13}{7}=2x\cdot7\) \(x-13=14x\)

To get the variable terms on the left side, subtract \(14x\) from both sides of the equation.

\(x-13-14x=14x-14x\) \(-13x-13=0\)

Now, add the constant of 13 to both sides.

\(-13x-13+13=0+13\) \(-13x=13\)

Then, divide both sides by –13.

\(\frac{-13x}{-13}=\frac{13}{-13}\) \(x=-1\)

\(3\cdot\frac{x+25}{3}=3(2x-30)\) \(x+25=6x-90\)

To get the variable terms on the left side, subtract \(6x\) from both sides of the equation.

\(x+25-6x=6x-90-6x\) \(-5x+25=-90\)

Now, subtract 25 from both sides.

\(-5x+25-25=-90-25\) \(-5x=-115\)

Then, divide both sides by –5.

\(\frac{-5x}{-5}=\frac{115}{-5}\) \(x=23\)

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## Solving Equations with Fractions

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

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!

We need to discuss the word term.

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

## Example 1 - Equations with Fractions

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

## Example 2 - Equations with Fractions with the Same Denominator

## Example 3 - Equations with Two Fractions with Different Denominators

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## Course: Algebra 1 > Unit 2

- Why we do the same thing to both sides: Variable on both sides
- Intro to equations with variables on both sides
- Equations with variables on both sides: 20-7x=6x-6

## Equation with variables on both sides: fractions

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Algebraic fractions are simply fractions with algebraic expressions on the top and/or bottom.

The video below shows you how to calculate algebraic fractions.

= 1(x + 6) + 4(x + 1) (x + 1)(x + 6)

= x + 6 + 4x + 4 (x + 1)(x + 6)

multiply both sides by x(x + 3): ∴ 10x(x + 3) - 2x(x + 3) = x(x + 3) (x + 3) x

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## IMAGES

## VIDEO

## COMMENTS

Equations involving algebraic fractions, where there are variables (letters) on the denominators of the fractions.

This makes the equation much harder to solve because we must use the rules of fraction addition and fraction multiplication to isolate the

Algebraic fractions are simply fractions with algebraic expressions either on the top, bottom or both. We treat them in the same way as we

How to solve equations including algebraic fractions · Convert each fraction so they all have a common denominator. · Multiply the equation throughout by the

To multiply algebraic fractions, first factor the numerators and denominators that are polynomials; then, reduce where possible. Multiply the remaining

In this second video we look at an alternative method to solve linear equations with algebraic fractions in. If you are confident, this method is more

are all algebraic fractions. ... This is a nice and simple one. We know that in order to solve an equation, we need to isolate our variable. Here

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

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

When adding or subtracting algebraic fractions, the first thing to do is to put them onto a common denominator (by cross multiplying). ... When solving equations