Problem 60 - Prealgebra

Question

PREREQUISITE SKILL Solve each equation. $$-3 y=-27$$

Step-by-Step Solution

Answer

The solution is \(y = 9\).

Step 1: Understand the Equation

We are given the equation \(-3y = -27\). Our goal is to isolate the variable \(y\). This will involve performing operations to both sides of this simple linear equation.

Step 2: Isolate the Variable

To isolate \(y\), divide both sides of the equation by \(-3\). This operation will solve for \(y\) by removing the coefficient next to it:\[ y = \frac{-27}{-3} \]

Step 3: Simplify the Expression

Next, simplify \(\frac{-27}{-3}\). Since both the numerator and the denominator are negative, the negatives cancel each other out, leaving:\[ y = 9 \]

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(y = 9\) back into the original equation:\(-3(9) = -27\),which simplifies to \(-27 = -27\), confirming that our solution is correct.

Key Concepts

Variable Isolation

When solving an equation like \(-3y = -27\), our main objective is to get the variable \(y\) by itself on one side of the equation. This technique is known as variable isolation. Variable isolation is a fundamental step in solving linear equations. Here, the variable \(y\) is multiplied by \(-3\), and our task is to remove the coefficient of \(y\) to isolate it. Let's go over the essential steps to achieve this:

• Recognize the variable you need to isolate—in this case, \(y\).
• Determine the coefficient affecting the variable—in this example, the coefficient is \(-3\).
• Use an operation that "undoes" what the coefficient is doing. Since \(y\) is multiplied by \(-3\), we divide both sides by \(-3\) to isolate \(y\).

By following these simple steps, we can efficiently isolate any variable in a linear equation.

Coefficient Manipulation

Manipulating the coefficient is a key technique when solving linear equations. In the equation \(-3y = -27\), the coefficient is the number \(-3\) that is multiplying our variable \(y\). Manipulating involves using the inverse operation to cancel out the coefficient.Here's the process of manipulating the coefficient:

• Identify the operation present with the coefficient—in this case, multiplication.
• Apply the inverse operation. For multiplication, the inverse operation is division.
• Divide both sides of the equation by the same number to maintain equality. For \(-3y = -27\), we divide both sides by \(-3\) to simplify and get \(y = 9\).

Remember, everything you do to one side of an equation, you must do to the other. This ensures that the equation remains balanced and correct.

Equation Verification

Once you find a solution, it's crucial to verify your answer. Verification confirms that no errors were made during calculation. For our solution \(y = 9\), we substitute this back into the original equation \(-3y = -27\) as follows:

• Replace \(y\) with your solution, in this case, \(9\).
• Calculate the left side: \(-3 \times 9 = -27\).
• Check if both sides of the equation are equal. If they are, the solution is verified. Here, \(-27 = -27\) shows both sides match, confirming the solution is correct.

Verification isn't just about checking; it reassures you of your understanding and accuracy in solving equations. Always make this your final step to ensure your solutions are reliable.