Solving Simultaneous Equations with Fractional Coefficients

Systems of Linear Equations in Two and Three Variables

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Staring at a system of equations where every variable is trapped inside a fraction can look intimidating. But you can eliminate every fraction in one single step without changing the solution.

A coefficient is the number multiplied by a variable, such as the

\frac{2}{3}

\frac{2}{3}x

. A system of simultaneous equations is a set of two or more equations sharing the same variables that must all be true at the same time.

To clear fractions, we find the least common denominator (LCD)—the smallest positive integer that each denominator divides into evenly—and multiply every term on both sides by it.

📊Interactive diagram

What happens when both equations in a system contain different fractions? Let's clear them systematically.

Clearing Denominators in a System

Consider the following simultaneous linear system:

\begin{cases} \frac{1}{2}x + \frac{2}{3}y = 4 \\[6pt] \frac{1}{4}x - \frac{1}{6}y = \frac{1}{2} \end{cases}

For Equation 1, the denominators are

2

and

3

, giving an LCD of

6

. For Equation 2, the denominators are

4

6

, and

2

, giving an LCD of

12

Multiplying Equation 1 by

6

gives

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

, which simplifies directly to

3x + 4y = 24

Multiplying Equation 2 by

12

gives

12\left(\frac{1}{4}x\right) - 12\left(\frac{1}{6}y\right) = 12\left(\frac{1}{2}\right)

, which simplifies to

3x - 2y = 6

📊Interactive diagram

Now that our equations contain only integers, how do we solve for

x

and

y

Solving Simultaneous Equations with Fractional Coefficients

Clearing Denominators in a System

Solving and Verifying the System