91Ó°ÊÓ

Linear equations are fundamental in algebra, representing relationships where each term is either a constant or the product of a constant and a single variable. To solve a linear equation, the goal is to find the value of the variable that makes the equation true. In our exercise, the equation given is \(-15n = 45\). Solving it means finding what number, when multiplied by \(-15\), gives \(45\). This process often involves inverse operations, such as addition or subtraction to undo constants or multiplication and division to undo coefficients of the variable.

In this specific problem, the multiplication principle is used. It states that you can multiply or divide both sides of an equation by the same non-zero number without changing the equality. This principle allows us to isolate the variable in the next step.

Isolating the variable is a crucial step in solving linear equations. The objective is to have the variable by itself on one side of the equation so you can easily see its solution. For the equation \(-15n = 45\), we want to isolate \(n\).

The most straightforward way to do this is by dividing both sides of the equation by \(-15\), the coefficient of \(n\). This division simplifies the equation to \(n = -3\).

Some tips for isolating the variable effectively include:

• Perform the same operation on both sides of the equation to maintain equality.
• Always divide by the coefficient, whether it is positive or negative, to solve for the exact value of the variable.
• Keep equations balanced at all times to ensure the solution is correct.

This step lays the groundwork for the final task: verifying the solution.

After finding a solution, it's important to verify it by substituting it back into the original equation. This ensures that the solution is accurate and complete.

For our problem, after solving for \(n\) and getting \(n = -3\), we substitute \(-3\) back into the original equation \(-15n = 45\) to check:

• Replace \(n\) with \(-3\): \(-15(-3) = 45\).
• Simplify the left side: \(45 = 45\).

The left side equals the right side, confirming that our solution \(n = -3\) is correct.

Checking solutions is a key habit in solving equations. It validates whether the steps taken were right and the solution is indeed valid, providing confidence in your mathematical skills.