91Ó°ÊÓ

Algebraic equations are mathematical statements that show the equality between two expressions. In this context, the equation given is \(\frac{r}{8} = -3\). The goal is to find the value of the variable that makes the equation true.
Students should understand that an equation can contain numbers, variables, and operations. For example:

• \(5x + 3 = 18\)
• \(2y - 4 = 10\)

In our problem, \(r\) is the variable we need to find. To solve algebraic equations, you perform operations to both sides to isolate the variable.

Isolating the variable means getting the variable on one side of the equation by itself. For the equation \(\frac{r}{8} = -3\), we need to isolate \(r\). To do this, we use inverse operations.
Inverse operations help 'undo' other operations. For example:

• To undo addition, you subtract.
• To undo multiplication, you divide.

In this case, \(r\) is divided by 8, so we do the opposite operation: multiplication. By multiplying both sides of the equation by 8, we isolate \(r\):
\[8 \times \frac{r}{8} = -3 \times 8\] which simplifies to \(r = -24\).

Multiplication is a commonly used operation when working with equations. To both sides of an equation, you can multiply or divide by the same number to keep the equation balanced.
For the equation \( \frac{r}{8} = -3\), multiplying both sides by 8 allows us to isolate the variable. This action is illustrated by:
\[8 \times \frac{r}{8} = -3 \times 8\]
Why does this work? Because 8 is the reciprocal of \(\frac{1}{8}\), effectively cancelling each other out on the left side:
\(\frac{8}{8} = 1\), and:
\(1 \times r = r\).
Thus, the left side simply becomes \(r\), and you get the equation:
\(r = -24\).
Checking our work, we substitute \(r\) back into the original equation to ensure it's correct.