Problem 15 - Prealgebra

Question

Solve each equation. Check your solution. $$5 n=-95$$

Step-by-Step Solution

Answer

The solution is \(n = -19\).

Step 1: Isolate the Variable

To solve the equation \(5n = -95\), the first step is to isolate \(n\). You do this by dividing both sides of the equation by 5, which is the coefficient of \(n\). So, \(n = \frac{-95}{5}\).

Step 2: Simplify the Expression

Now, simplify the expression on the right-hand side: \(\frac{-95}{5} = -19\). Therefore, \(n = -19\).

Step 3: Verify the Solution

To ensure the solution is correct, substitute \(n = -19\) back into the original equation. The original equation is \(5n = -95\). Replacing \(n\) with \(-19\), we have \(5(-19) = -95\). Calculate \(5 \times -19\), which results in \(-95\). Since both sides of the equation are equal, the solution is verified.

Key Concepts

Solving Equations

When faced with an equation like \(5n = -95\), the goal is to determine the value of the unknown variable, in this case, \(n\). Solving equations involves finding the value of the variable that makes the equation true. Here are some basic steps to follow:

• Identify the equation and the variable you need to solve for. In our example, the variable is \(n\).
• Decide on the operations needed to isolate the variable, which will help in finding its value.

By solving this equation correctly, you ensure the balance between both sides of the equation is maintained. Keeping the equation balanced is essential for arriving at a correct solution.

Isolating the Variable

Isolating the variable is a critical step in solving equations. The goal is to have the variable by itself on one side of the equation, with a numerical value on the other side. This often requires basic arithmetic operations such as addition, subtraction, multiplication, or division.

In our example, \(5n = -95\), the variable \(n\) is multiplied by 5. To isolate \(n\), divide both sides by 5:

• Divide both the left side and the right side of the equation by the same number, here it is 5.
• This leaves us with \(n = \frac{-95}{5}\).

After performing the division, we find \(n = -19\). The isolation of the variable is complete when \(n\) stands alone, indicating what value it must take to satisfy the original equation.

Verifying Solutions

After finding the potential solution to an equation (in this case, \(n = -19\)), it is important to verify that the solution is correct. Verification involves substituting the value back into the original equation to ensure both sides are equal. Here's how you can do it:

• Take the value you obtained for the variable and substitute it back into the original equation.
• For our problem, replace \(n\) with \(-19\) in \(5n = -95\) to get \(5(-19)\).
• Calculate the result on the left side: \(5 \times -19 = -95\).

Since both sides of the equation equal \(-95\), the solution \(n = -19\) is verified as correct. Always remember, verifying solutions helps confirm the accuracy of your findings and strengthens your understanding of solving equations.