Solving with Algebraic Fractions
Neil Trivedi
Teacher
Contents
Solving with Algebraic Fractions
Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. When solving equations involving algebraic fractions such as
our aim is to remove the denominators so that the equation becomes easier to solve.
We can eliminate the denominators by multiplying every term in the equation by the lowest common multiple of the denominators.
In the above example, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by
Simplifying the left-hand side by collecting like terms,
Dividing both sides by
Solving with Numeric Denominators
Example 1:
Solve for :
a)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by to eliminate the denominators.
Step 2: Solve for
Simplifying the left-hand side by collecting like terms,
Dividing both sides by
b)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by to eliminate the denominators.
Note: When we see more than one term in a fraction such as and we put a bracket around them before cancelling. A bracket protects them and it becomes one term.
Step 2: Solve for
First, we simplify the left-hand side by expanding the brackets.
Collecting like terms,
Subtracting from both sides,
Dividing both sides by
c)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by to eliminate the denominators.
Step 2: Solve for
First, we move the to the right-hand side and the to the left-hand side so that all the
terms are on one side and all the constants are on the other.
Dividing both sides by
Solving with Algebraic Denominators
We will now solve equations involving algebraic fractions where the denominator contains an algebraic expression. The process is similar to when the denominators are numerical.
First, we find the lowest common multiple of the denominators. Then, we multiply every term in the equation by this lowest common multiple to eliminate the denominators. Finally, we solve the resulting equation for
Remember to put brackets around any numerator or denominator with more than one term.
Example 2:
Solve for
a)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by to eliminate the denominators.
Step 2: Solve for
First, we expand the brackets on both sides.
Adding to both sides and subtracting from both sides,
b)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the only denominator is So, we multiply every term in the equation by to eliminate that denominator.
Step 2: Solve for
First, we expand the brackets on the right-hand side.
Adding to both sides and subtracting from both sides,
Dividing both sides by
Example 3:
Solve for
a)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the only denominator is So, we multiply every term in the equation by to eliminate that denominator.
Step 2: Solve for
First, we expand the brackets on both sides.
We now have a quadratic equation so, we bring all terms to one side. We subtract from both sides and add to both sides to bring all terms to the left-hand side.
Next, we factorise the quadratic equation. For more information on how to factorise quadratics, please read our “Factorising Quadratics” note.
Setting each bracket equal to we have
or
Solving each linear equation for
or
b)
Step 1: Find the lowest common multiple of the denominators and multiply every term in the equation by it.
Here, the denominators are and The lowest common multiple of and is so we multiply every term in the equation by to eliminate the denominators.
Step 2: Form a quadratic equation in terms of with one side equal to
First, we expand the brackets on both sides.
Expanding the right-hand side and collecting like terms on the left-hand side,
We now bring all terms to one side. We subtract and from both sides.
Step 3: Solve the quadratic equation for
The quadratic equation can be factorised using the OI-OI method, coined by NeilDoesMaths. Please read our “Factorising Quadratics” note to revise this method if needed.
Once we factorise the quadratic, we get
Setting each bracket equal to we have
or
Solving each linear equation for
or
Challenging Questions