Problem 14 - Prealgebra

Question

Solve each equation. Check your solution. $$-32=4 y$$

Step-by-Step Solution

Answer

The solution is \(y = -8\).

Step 1: Isolate the Variable

Start with the equation \(-32 = 4y\). To isolate \(y\), divide both sides of the equation by 4. This yields \(y = \frac{-32}{4}\).

Step 2: Simplify the Expression

Now simplify the division on the right side. Calculate \(-32 \div 4\), which equals \(-8\). So, \(y = -8\).

Step 3: Verify the Solution

Substitute \(y = -8\) back into the original equation to check the solution. The original equation is \(-32 = 4y\). Replace \(y\) with \(-8\): \(-32 = 4(-8)\). Calculate \(4 \times (-8)\) to verify: it equals \(-32\), which confirms the solution is correct.

Key Concepts

Isolating the Variable

To solve equations like \(-32 = 4y\), the first step is to isolate the variable. In our equation, the variable is \(y\), multiplied by 4. Isolating the variable means getting \(y\) on one side of the equation by itself.

Here's how you do it:

• Identify the operation acting on the variable (in this case, multiplication by 4).
• To undo the multiplication, do the opposite operation. Divide both sides of the equation by 4 to balance it out.

Performing the division gives us:

• \(y = \frac{-32}{4}\)

Now, \(y\) is by itself on one side. This is a crucial step because it allows you to see clearly what value \(y\) has to be.

Simplifying the Expression

After isolating the variable, the next step is simplifying the expression that results from the division. When you have \(y = \frac{-32}{4}\), you need to perform the arithmetic operation to achieve the simplest form of the solution.

Let's break it down:

• Calculate \(-32 \div 4\).
• Divide \(-32\), which is negative, by \(4\), which is positive, to get \(-8\).

So, the simplified expression gives you:\

• \(y = -8\)

This is the simplest and final form of your solution for \(y\). Breaking down each operation ensures a clear understanding of how we arrived at \(-8\).

Verifying the Solution

Verification is the essential final step to ensure the solution is correct. Even if you feel confident, always double-check your results. For our solution \(y = -8\), we plug it back into the original equation \(-32 = 4y\) to verify.

Here's what you should do:

• Substitute \(y\) with \(-8\) in the original equation.
• It looks like this: \(-32 = 4(-8)\).
• Calculate \(4 \times (-8) = -32\).

The left side \(-32\) equals the right side \(-32\), so the solution \(y = -8\) is indeed correct. Verification confirms that your calculations and logic are sound, offering peace of mind and guaranteeing the accuracy of your results.