91Ó°ÊÓ

Linear equations are a fundamental component of algebra and represent relationships where the highest power of the variable is 1. In these equations, each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in one variable, like the one in our exercise, is typically written as:

• x + b = c

This type of equation has a straightforward solution process, as demonstrated in the original exercise. Here, our task was to solve the equation \( \frac{d}{-7} = -3 \). This means we needed to find the value of \(d\) that would make the equation true. When working with linear equations like this one, the aim is to isolate the variable, making it easier to determine its value through algebraic manipulation.

Algebraic manipulation is the process required to rearrange and simplify equations to isolate and solve for unknown values. In our solution, we performed straightforward algebraic manipulation to solve the equation \( \frac{d}{-7} = -3 \). This involved using multiplication to eliminate the fraction:

• The equation started as \( \frac{d}{-7} = -3 \).
• By multiplying both sides by -7, we eliminated the denominator on the left side.
• This step effectively isolated \(d\) and simplified to \(d = 21\).

These types of transformations are crucial in solving linear equations because they maintain the balance of the equation—whatever you do to one side, you must do to the other.

Verifying your solution is an essential part of solving any equation. It ensures that the solution is correct and that no mistakes were made during the manipulation process. After finding \(d = 21\), we verified our work by substituting the solution back into the original equation:

• We replaced \(d\) with 21 in the equation \( \frac{d}{-7} = -3 \).
• This gave us the expression \(\frac{21}{-7}\).
• Calculating this, we found that \(21 \div -7 = -3\), which matches the original equation.

Since both sides of the equation were equal after substitution, our solution was confirmed. Equation verification is a critical final step as it provides proof of accuracy and reinforces understanding of the process.