91Ó°ÊÓ

The concept of isolating the variable is crucial in solving linear equations. When we isolate the variable, we aim to get the unknown (in this case, x) all by itself on one side of the equation. This helps us easily see what the variable equals.

In the given equation \(\frac{x}{7} = -5\), we want to isolate x. Initially, x is being divided by 7. To counterbalance this and isolate x, we need to do the opposite of dividing, which is multiplying.

When we multiply both sides of the equation by 7, we effectively cancel out the division by 7 on the left side. The equation transforms into \[ x = -35 \]. Now x is isolated, meaning we have successfully found the value of x that satisfies the equation.

One of the key techniques used in solving equations is the multiplication property of equality. This principle states that when you multiply both sides of an equation by the same number, the two sides remain equal.

This property was used in Step 2 of our solution. We started with the equation \(\frac{x}{7} = -5\). To clear the fraction and isolate x, we multiplied both sides by 7:

• On the left side: \( 7 \times \frac{x}{7} = x \) because the 7s cancel out.
• On the right side: \( -5 \times 7 = -35 \).

By applying this property, the equation becomes \( x = -35 \), where x is now isolated with its value determined.

Verifying solutions is an essential step to ensure the correctness of your answer. After finding the solution, we substitute it back into the original equation to check if it holds true.

For the equation \(\frac{x}{7} = -5 \) and the solution \( x = -35 \), the verification step is straightforward:

• Substitute \( x = -35 \) back into the original equation: \( \frac{-35}{7} \).
• Simplify the left side: \( \frac{-35}{7} = -5 \).
• Both sides of the equation are equal (\( -5 = -5 \)) which confirms that the solution \( x = -35 \) is indeed correct.

Verifying solutions gives you confidence that your solution is accurate and helps in identifying any mistakes made during the solving process. This step is crucial for building a strong foundation in algebra.