How to Solve Linear Equations with Fractions in Year 10 Math

Solving linear equations with fractions might seem hard at first, but it can be easy and fun if you know how to do it. Let’s look at some simple ways to solve these equations step by step.

Understanding the Basics

A linear equation with fractions can look like this:

$\frac{x}{3} + 2 = \frac{5}{2}$

We want to find the value of $x$. The first step is to get rid of the fractions so the equation is easier to manage.

Step 1: Get Rid of the Fractions

One good way to eliminate fractions is to find the least common denominator (LCD).

In our example, the denominators are 3 and 2. The LCD is 6.

Now, we can multiply every part of the equation by 6:

$6 \left(\frac{x}{3}\right) + 6(2) = 6\left(\frac{5}{2}\right)$

This simplifies to:

$2x + 12 = 15$

Step 2: Solve the New Equation

Now that there are no fractions, we can solve the equation.

First, subtract 12 from both sides:

$2x = 15 - 12$

Which simplifies to:

$2x = 3$

Next, divide both sides by 2 to get $x$ by itself:

$x = \frac{3}{2}$

So, the answer is $x = 1.5$.

Another Example

Let's try a more complicated equation:

$\frac{2x - 1}{4} = \frac{x + 3}{2}$

First, we find the LCD, which here is 4. We’ll multiply everything by 4:

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

This simplifies to:

$2(2x - 1) = 2(x + 3)$

Now, let’s expand both sides:

$4x - 2 = 2x + 6$

Next, we solve for $x$ by subtracting $2x$ from both sides:

$2x - 2 = 6$

Then, add 2 to both sides:

$2x = 8$

Finally, divide by 2:

$x = 4$

Conclusion

To solve linear equations with fractions easily:

1. Find the least common denominator.
2. Multiply the whole equation to remove fractions.
3. Simplify and solve for the variable.

With practice, dealing with fractions will become a breeze in your math problems!