Problem 27 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{v}{-11}=-132$$

Step-by-Step Solution

Answer

The solution is \( v = 1452 \).

Step 1: Isolate the Variable

To solve the equation \( \frac{v}{-11} = -132 \), you need to get \( v \) by itself on one side of the equation. You can do this by multiplying both sides of the equation by \(-11\). This will cancel out the \(-11\) in the denominator on the left side of the equation, resulting in: \[ v = -132 \times (-11) \]

Step 2: Calculate the Value of the Variable

Now, calculate the right side of the equation. Multiply \(-132\) by \(-11\): \[ v = 1452 \]Negative times negative gives a positive.

Step 3: Verify the Solution

Substitute \( v = 1452 \) back into the original equation to ensure the solution is correct:\[ \frac{1452}{-11} = -132 \]Calculate the left side: \[ 1452 \div -11 = -132 \]Since both sides are equal, the solution is verified.

Key Concepts

Isolate the Variable

Solving equations is like solving a puzzle where you aim to figure out the unknown value by yourself. The first major step in solving any equation is to isolate the variable. This simply means you want the variable by itself on one side of the equation. In the given equation, \( \frac{v}{-11} = -132 \), we’d like to separate \( v \) from the rest of the equation. To do this effectively, you perform the opposite mathematical operation on both sides without changing the equation's balance.

• Here, \( v \) is divided by \(-11\). To get \( v \) alone, you can do the opposite: multiply both sides by \(-11\).
• This cancels out the \(-11\) on the left side, leaving \( v \) by itself.

This crucial step of isolation often simplifies the equation to a much more manageable state.

Substitution Method

After isolating the variable and determining its value, it’s essential to check that this solution satisfies the original equation. This brings us to the substitution method. This step checks your work, ensuring no miscalculations were made. For our equation, once \( v \) is calculated to be \( 1452 \), substitute this value back into the original equation to verify:
\( \frac{1452}{-11} \).

• If your computations are correct, simplifying this should give you \(-132\).
• If both sides of the equation are equal, you have verified the solution is correct.

The substitution method is a robust way to confirm your findings and avoid small errors.

Negative Numbers

Handling negative numbers can be tricky but understanding a few basic principles makes it easier. When you multiply or divide two negative numbers, the result turns out to be positive. In the equation \( \frac{v}{-11} = -132 \), you multiply both sides by \(-11\) and get the expression:
\( -132 \times (-11) \).

• When carrying out this operation, remember that \(-\times-\) results in a positive.
• This is why \( v \) equals \( 1452 \), a positive number.

Recognizing these sign rules helps prevent common mistakes when solving equations involving negative numbers.

Verify the Solution

In mathematics, verifying your solution is like double-checking your work to make sure everything makes sense. Verification instills confidence in your answer.After you've found a solution, plug it back into the original equation. Our distinguished analysis with solving \( \frac{v}{-11} = -132 \) yielded \( v = 1452 \). Substituting back gives us:
\( \frac{1452}{-11} = -132 \).

• Perform the division to make sure both sides of the equation match.
• If they do match, you can confidently say your solution is correct.

This process of verification is crucial, especially in more complex equations, as it helps catch errors early.