91Ó°ÊÓ

To solve an equation, we often need to 'isolate the variable'. This means getting the variable we are solving for, such as \( x \), by itself on one side of the equation. In the equation \( -\frac{1}{4} x = 3 \), our goal is to move everything else away from \( x \). By doing this, we can find the value of \( x \). This requires us to perform operations that will cancel out any other numbers or coefficients attached to \( x \).

For example, in this problem, \( x \) is multiplied by \( -\frac{1}{4} \). To isolate \( x \), we need to do the opposite of multiplying by \( -\frac{1}{4} \), which is multiplying by \( -4 \). This way, the fraction on \( x \) is removed, and \( x \) stands alone.

Multiplying fractions might look tricky, but it's straightforward. In this problem, we're dealing with the fraction \( -\frac{1}{4} \) that multiplies \( x \). To eliminate this fraction, we need to multiply both sides of the equation by its reciprocal. The reciprocal of \( -\frac{1}{4} \) is \( -4 \).

Here's why: When you multiply \( -\frac{1}{4} \) by \( -4 \), the fractions cancel each other out, simplifying the equation. So, multiplying both sides of \( -\frac{1}{4} x = 3 \) by \( -4 \) gives us:

• On the left side: \( -4 \times -\frac{1}{4} x = x \)
• On the right side: \( -4 \times 3 = -12 \)

Therefore, multiplying fractions correctly helps us in simplifying and solving equations.

Simplifying equations is about making them easier to solve. After isolating the variable and performing appropriate operations, we get a much simpler equation. This reduces the steps needed to find the solution.
In our example, after multiplying both sides of \( -\frac{1}{4} x = 3 \) by \( -4 \), we simplified the left side, leaving us with:

• \( x = -12 \)

By simplifying, we see clearly that \( x = -12 \). There are no further steps needed because the equation is now in its simplest form. Simplification helps us avoid unnecessary complexity and easily see the solution.