91Ó°ÊÓ

When solving linear equations, the main goal is to find the value of the unknown variable. This process is called isolating the variable.
To isolate a variable means to get the variable by itself on one side of the equation.
You usually do this by performing the same operation on both sides of the equation, such as addition, subtraction, multiplication, or division.

In our example \( \frac{1}{4}x = -12 \), we need to isolate \( x \).
Because \( x \) is multiplied by \( \frac{1}{4} \), we perform the inverse operation, which is multiplication by 4, to both sides:
\[ \frac{1}{4} x \times 4 = -12 \times 4 \]
This allows us to cancel out the fraction and gain \( x \) by itself, simplifying the equation to \( x = -48 \).

Multiplication is one of the core operations in algebra and is essential when solving equations.
In this problem, when we multiply both sides by 4, it's important to perform multiplication accurately.

Considering our example, after isolating the variable step:
\[ x = -12 \times 4 \]
Here, we multiply \( -12 \) by \( 4 \). Let’s break this down:
- Think of it as \( -12 \) added 4 times: \( -12 + -12 + -12 + -12 \)
- This results in \( -48 \).

Negative times positive always results in a negative number.
Therefore, \( -12 \times 4 = -48 \).

Fractions represent parts of a whole and are written in the form \( \frac{numerator}{denominator} \).
In this problem, \( \frac{1}{4} x = -12 \), the fraction \( \frac{1}{4} \) means dividing \( x \) by 4.

To cancel out the fraction and isolate the variable, we do the inverse operation, which is multiplication.
Multiplying both sides of the equation by the denominator can clear the fraction:
\[ \frac{1}{4} x \times 4 = -12 \times 4 \]
Simplifying this yields \( x = -48 \).
Recognizing how to properly manipulate fractions is vital in algebra to balance and solve equations.