91Ó°ÊÓ

Dealing with fraction coefficients in equations can be a bit tricky, but it becomes easier with practice.
A fractional coefficient is simply a fraction that multiplies the variable.
In the given exercise, the coefficient of the variable x is \(\frac{3}{4}\).
When solving such equations, you should first get comfortable with adding, subtracting, multiplying, and dividing fractions.
Let's break it down further:

• \textbf{Addition}: To add fractions, ensure they have a common denominator before combining them.
  For example, adding \(\frac{1}{4}\) and \(\frac{1}{2}\) gives \(\frac{1}{4} + \frac{2}{4} = \frac{3}{4}\).
• \textbf{Multiplication}: Multiply the numerators together and the denominators together.
  For instance, multiplying \( \frac{3}{4} \times \frac{4}{3} \) is straightforward: you get \( \frac{3 \times 4}{4 \times 3} = 1 \).
• \textbf{Division}: Dividing fractions is the same as multiplying by the reciprocal.
  This will be explored further in the section on reciprocals.

Being comfortable with these operations will help you solve equations with fraction coefficients more smoothly.

The goal in solving an equation is usually to isolate the variable.
This means getting the variable by itself on one side of the equation.
The first step is to eliminate any constants or other terms around the variable.
In the original problem, you need to deal with \( \frac{3}{4}x - \frac{1}{2} = \frac{1}{4} \).
To isolate x, start by moving the constant term \( - \frac{1}{2} \) from the left side.
This can be achieved by adding \( \frac{1}{2} \) to both sides:

• \( \frac{3}{4}x - \frac{1}{2} + \frac{1}{2} = \frac{1}{4} + \frac{1}{2} \)

On simplifying, we get:
\( \frac{3}{4}x = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \).
Now, it's easier to move to the next step.

Always ensure the side with the variable contains only the variable term before moving on.

The reciprocal of a fraction flips its numerator and denominator.
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
Reciprocals are important when we need to cancel out multiplication by a fraction.
To isolate x in the equation \( \frac{3}{4}x = \frac{3}{4} \), multiply both sides by the reciprocal of \( \frac{3}{4} \):

• Applying \( \frac{3}{4} \times \frac{4}{3} \) to both sides:
• \( \frac{3}{4}x \times \frac{4}{3} = \frac{3}{4} \times \frac{4}{3} \)

This simplifies to:
\( x = 1 \).
Multiplying by the reciprocal effectively removes the fraction, leaving the variable isolated.
Remember, using reciprocals is a helpful step when dealing with equations containing fractional coefficients.