Problem 19 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ -3 y=-42 $$

Step-by-Step Solution

Answer

The solution is \( y = 14 \).

Step 1: Isolate the Variable

To solve for \( y \), we need to isolate it by dividing both sides of the equation \( -3y = -42 \) by \( -3 \). This is done to cancel out the \( -3 \) that is being multiplied by \( y \). Thus, we have: \[ y = \frac{-42}{-3} \]

Step 2: Simplify the Division

Now, simplify the division on the right-hand side of the equation: \[ y = \frac{-42}{-3} = 14 \] This means \( y = 14 \).

Step 3: Check the Solution

Substitute \( y = 14 \) back into the original equation to verify: \[ -3(14) = -42 \] Calculating the left side gives us \(-42\), which matches the right side of the equation, confirming our solution is correct.

Key Concepts

Isolating Variables

When solving linear equations, isolating the variable is a crucial step. This means getting the unknown (in this case, the variable \( y \)) on one side of the equation by itself. We aim to have \( y \) on its own on one side to make it easier to identify its value. Consider the equation, \(-3y = -42\). Here, \(-3\) is multiplied by \( y \). To "undo" this multiplication, we do the opposite operation, which is division. Thus, we divide both sides of the equation by \(-3\):
\( y = \frac{-42}{-3} \).
This cancels out the \(-3\) on the left-hand side, leaving us with just \( y \), now on its own. The logic is simple: whatever operation you perform on one side, you must do it on the other to maintain balance.

Simplifying Equations

After isolating the variable, the next step is to simplify the equation. Simplifying involves performing any arithmetic operations to make the equation as straightforward as possible. In our example, after isolating \( y \), we get \( y = \frac{-42}{-3} \).
Here, simplifying this fraction means dividing \(-42\) by \(-3\).

• First, remember that dividing a negative by a negative gives a positive result.
• Second, perform the division: \(-42 \div -3 = 14\).

Now, we've simplified the right-hand side to just \(14\), revealing that \( y = 14 \). This simplification step helps to see the solution clearly and ensures that we've correctly calculated the value of the variable.

Checking Solutions

Once we have found a solution, it's important to verify that it is correct by checking it. This means substituting the solution back into the original equation. In this example, we found \( y = 14 \).
To check:
Substitute \( y = 14 \) back into the original equation \(-3y = -42\):

• Replace \( y \) with \( 14 \): \(-3(14)\).
• Calculate the left side: \(-3 \times 14 = -42\).

The left side now equals \(-42\), which is the same as the right side of the original equation. This confirms that our solution \( y = 14 \) is indeed correct. Checking your work is crucial as it catches any possible mistakes and gives confidence that the solution is valid.