Problem 17 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$-8 y=64$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -8\).

Step 1: Understand the Equation

We start with the equation \(-8y = 64\). The goal is to isolate the variable \(y\) by using the multiplication property of equality.

Step 2: Apply the Multiplication Property of Equality

To isolate \(y\), divide both sides of the equation by \(-8\), since \(y\) is multiplied by \(-8\). Thus, we have: \[y = \frac{64}{-8}\]

Step 3: Simplify the Expression

Simplify the fraction \(\frac{64}{-8}\) by performing the division: \[y = -8\]

Step 4: Verify the Solution

Substitute \(y = -8\) back into the original equation to check if it holds true: \[-8(-8) = 64\] This simplifies to \(64 = 64\), which confirms our solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that satisfies the given equation. It requires a systematic approach to transform the equation into a simpler form, enabling us to identify the variable's value easily. Each step must adhere to the property of equality, ensuring anything done to one side of the equation is equally applied to the other. This maintains balance in the equation and ensures that the solution is valid.

To effectively solve a linear equation, you can follow these general steps:

• Identify the operation affecting the variable (e.g., multiplication, division, addition, or subtraction).
• Choose the opposite operation to eliminate it and focus on isolating the variable.
• Simplify both sides as needed, maintaining balance throughout the process.
• Recheck each step to prevent errors from creeping into the calculations.

Isolate Variable

Isolating the variable is a critical step when solving linear equations. This term refers to the process of manipulating the equation so that the variable stands alone on one side of the equation. In the exercise, -8 is multiplying the variable y. Our goal is to remove this multiplication, which requires us to use its inverse operation.

Steps to Isolate the Variable:

• Identify what is being done to the variable. In this case, y is being multiplied by -8.
• Apply the inverse operation. Here, dividing both sides of the equation by -8 cancels out the multiplication. This yields: \(y = \frac{64}{-8}\).
• Simplify the result to isolate the variable completely. Performing the division gives us \(y = -8\).

By isolating the variable, you translate the problem into a straightforward solution, highlighting the precise value the variable represents in the context of the equation.

Verify Solution

Verifying a solution is always an important step in problem-solving. After finding the value of the variable, check to ensure it satisfies the original equation, confirming the solution is accurate. We substitute the found value back into the initial equation to test its validity.

Steps to Verify a Solution:

• Substitute the solution back into the original equation. For example, substitute y = -8 into \-8y = 64\.
• Calculate both sides of the equation after substitution; ensure that both sides equal the same value.
• If they do, the solution is correct. In our case: \(-8(-8) = 64\) simplifies to \(64 = 64\), confirming the correctness of the result.

Verification prevents errors from slipping through and boosts confidence in the obtained solution. It's a habit that reinforces sound mathematical practices, ensuring each solution is reliable and valid.