# Fractions in Expressions and Equations: Rules (2023)

Algebraic expressions in mathematics consist of constants and variables related by algebraic operations (additions, subtractions, multiplications, and divisions). A fraction is the division of two expressions: for example $\frac{1}{2},\frac{x+1}{3},\frac{xy}{x+y}$ are fractions of expressions containing constants and / or variables.

An equation is a mathematical statement that consists of an equal symbol between two algebraic expressions. For example, $\frac{1}{2}+x=2x+1$ is an equation.

Solving an equation means finding the value of the variable that makes the expressions on the left-hand side and the right-hand side equal; this value is called the solution of the equation. For example, the solution of the equation $\frac{1}{2}+x=2x+1$ is $x=-\frac{1}{2}$: if you plug in this value on both sides of the equation you get $0=0$.

This article deals with the presence of fractions in expressions and equations; such equations are called (unsurprisingly) fractional equations.

The primary step in dealing with fractional equations is to eliminate the fractions from them. Let's dive in and see how to achieve this!

## What are Terms, Coefficients, Variables, and Constants?

• Terms are the building blocks of algebraic expressions: in an expression, each term is separated by a plus (+) sign or a minus (-) sign. In the equation $3x+4=10$, 3x and 4 are the terms of the expression on the left-hand side; and 10 is the term of the expression on the right-hand side.
• Coefficients are the values that multiply the variables in either an expression or an equation. Given the expression $\frac{3}{4}x+4$, the coefficient is that number that multiplies the x, which is$\frac{3}{4}$.
• Variables are the letters in expressions and equations that are used to represent unknown quantities. When you have an expression given as $\frac{3}{2}x+\frac{1}{2}y$, x and y both identify as variables.
• Constants are the numbers in expressions and equations that do not change. In the equation $2x-4y+4=0$ for example, 4 stands to be the constant.

Figure 1. Terms in algebra, siyavula.com

## How to Solve Expressions with Fractions: Step-by-Step Examples

If you are dealing with fractions in expressions, it is easiest to add and subtract them when there are common denominators. This means that we will find the equivalent fraction of the fractions involved by finding the Lowest Common Divisor (LCD) for the denominators of the terms.

Simplify $\frac{3x}{5}+\frac{x}{4}$

Solution:

What we will do here is to find a common denominator for the two terms so they can be added. Firstly, we will have to find the LCD for the denominators of the two fractions, 5 and 4. The lowest common divisor for both numbers will be 20. Now we will find the equivalent fraction for both.

If the denominator for the first fraction should now be 20, that means we probably multiplied the initial denominator, 5 by 4. This would mean we will have to multiply the numerator also by 4 to have an equivalent fraction.

$\frac{4}{4}\left(\frac{3x}{5}\right)=\frac{12x}{20}$

We will do the same for the second fraction. 20 as a denominator also means that we must have multiplied 4 (as the denominator) by 5 to have 20 as the new denominator of the equivalent fraction. This means we will have to multiply the numerator also by 5.

$\frac{5}{5}\left(\frac{x}{4}\right)=\frac{5x}{20}$

(Video) Let’s Solve An Equation with Fractions….Step-by-Step….

We now have our new expression:

$\frac{12x}{20}+\frac{5x}{20}$

This becomes a lot easier to solve since all we have to do is add the numerators and maintain the denominators.

$\frac{12x}{20}+\frac{5x}{20}=\frac{17x}{20}$

Since this cannot be simplified any further, we will leave it at that.

### Solve Expressions with Fractions by Factorizing and Grouping

There could be more complicated problems where we may have to use a couple of techniques like factorizing and grouping. In these situations, one needs to be very careful about what particularly a term is, and when its components are being divided. Let us look at the example below:

Simplify $\frac{ax-b+x-ab}{a{x}^{2}-abx}$

Solution:

Since we cannot cancel anything in this current expression, we may want to factorize to see what we can make of the situation. We will first group like terms in the numerator by rearranging so that terms containing x will be close together and terms containing b will also be close.

$\frac{ax-b+ x-ab}{a{x}^{2}-abx}$

$\frac{ax+x-b-ab}{a{x}^{2}-abx}$

We will now factorize. The common factor in the first two terms in the numerator is x. That can be factored out. The common factor in the last two terms of the numerator is -b, and that can also be factored out.

$\frac{x\left(a+1\right)-b\left(1+a\right)}{a{x}^{2}-abx}$

The whole idea of factorizing here is building a common bracket so one can be taken out. Here we have $\left(a+1\right)$ and $\left(1+a\right)$. Considering the commutative property of addition, both brackets are the same.

This will leave us with:

$\frac{\left(x-b\right)\left(a+1\right)}{a{x}^{2}-abx}$

Now, we will factorize the denominator immediately. As ax appears common in both terms, that is what will be factored out.

$\frac{\left(x-b\right)\left(a+1\right)}{ax\left(x-b\right)}$

(Video) Solve Equations with Fractions| Expressions & Equations | Grade 7

We are now left with a situation where we can freely cancel out. $\left(x-b\right)$ in the denominator will cancel out $\left(x-b\right)$ in the numerator. This is true if $x$ is different from $b$.

$\frac{\left(a+1\right)}{ax}$

This is the simplest form we can get from this expression.

## How to Solve Equations with Fractions: Step-by-Step Examples

As mentioned earlier, the thing one should have their eyes on when dealing with equations involving fractions is to try to eliminate the fraction first. You should multiply all of the terms on both sides of the equation by the fraction's denominator.

If we were given the equation $5x+\frac{1}{2}=12$, we would first multiply the equation (which is technically also each term of the equation) by 2.

Solution:

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

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

After multiplying by 2, the fraction will cancel out.

$10x+1=24$

We will now rearrange the equation to put like terms on different sides of the equation.

$10x=24-1$

$10x=23$

Divide both sides by 10

$\frac{10x}{10}=\frac{23}{10}$

$x= 2.3$

To check that this is indeed the solution of the equation you need to substitute the value of x back into the original equation:

(Video) Simplifying Exponents With Fractions, Variables, Negative Exponents, Multiplication & Division, Math

$5\left(2.3\right)+\frac{1}{2}=12$

$11.5+\frac{1}{2}=12$

$12=12$

Solve$\frac{3}{2}x+\frac{1}{2}\left(x-4\right)=6$

Solution:

An equation with two fractions with the same denominator will have its terms being multiplied with the denominator, as mentioned earlier.

$2\left(\frac{3}{2}x\right)+2\left(\frac{1}{2}\left(x-4\right)\right)=2\left(6\right)$

$3x+x-4=12$

$4x-4=12$

Like terms will now be grouped from this point.

$4x=12+4$

$4x=16$

Divide both sides by 4

$\frac{4x}{4}=\frac{16}{4}$

$x=4$

To evaluate this, you would need to substitute the value of x back into the original equation.

$\frac{3}{2}\left(4\right)+\frac{1}{2}\left(4-4\right)=6$

$\left(3×2\right)+\frac{1}{2}\left(0\right)=6$

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$6+0=6$

$6=6$

Solve$\frac{1}{4}x+\frac{3}{2}\left(2x-1\right)=2$

Solution:

Our example is quite different from the usual here. Since we have two fractions with different denominators, we will find the LCM for both and multiply that with the equation. The LCM is 4, so

$4\left(\frac{1}{4}x\right)+4\left(\frac{3}{2}\left(2x-1\right)\right)=4\left(2\right)$

$1x+2\left(3\left(2x-1\right)\right)=4\left(2\right)$

We will now expand what is in the parenthesis:

$1x+2\left(6x-3\right)=8$

$1x+12x-6=8$

Group like terms:

$12x+1x=8+6$

$13x=14$

Divide both sides by 13:

$\frac{13x}{13}=\frac{14}{13}$

$x=\frac{14}{13}$

To evaluate this, you would need to substitute the value of x back into the original equation.

$\frac{1}{4}\left(\frac{14}{13}\right)+\frac{3}{2}\left(2\left(\frac{14}{13}\right)-1\right)=2$

(Video) Fractions Basic Introduction - Adding, Subtracting, Multiplying & Dividing Fractions

$2=2$

## Fractions in Expressions and Equations - Key takeaways

• The left-hand side and right-hand side of an equation must remain equal when operating on them.
• The primary step in dealing with fractional equations is to eliminate the fractions from them.
• When you have an equation with two fractions and different denominators, find the LCM for both numbers.

## Videos

1. 08 - Learn to Solve Fractional Equations in Algebra, Part 1
(Math and Science)
2. Simplifying rational expressions introduction | Algebra II | Khan Academy
3. Solving Rational Equations
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4. Solve an equation for x by clearing fractions with multiple steps
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5. Strategies to Solve Multi Step Linear Equations with Fractions
(Anil Kumar)
6. Algebra - Simplifying Complex Fractions
(The Organic Chemistry Tutor)

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