Problem 6 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(-y=8\)

Step-by-Step Solution

Answer

The solution is \(y = -8\). It checks out when substituted back into the equation.

Step 1: Understand the Equation

We are given the equation \(-y = 8\). The goal is to solve for \(y\). This equation suggests that the opposite of \(y\) equals 8.

Step 2: Isolate the Variable

To solve for \(y\), we need to get \(y\) by itself. We do this by multiplying both sides of the equation by \(-1\). Thus, \(-1(-y) = -1(8)\).

Step 3: Simplification

When simplifying, we have \(y = -8\) because \(-1 \times -y = y\) and \(-1 \times 8 = -8\).

Step 4: Check the Solution

To ensure our solution is correct, substitute \(y = -8\) back into the original equation. The equation \(-y = 8\) becomes \(-(-8) = 8\), which simplifies to \(8 = 8\), confirming that our solution is correct.

Key Concepts

Understanding Algebraic Manipulation

Solving linear equations often requires using algebraic manipulation, a powerful tool that helps us rearrange equations and find solutions. Think of it as a method to "clean up" equations by performing operations that maintain equality. These operations can include:

• Adding or subtracting the same number from both sides of the equation
• Multiplying or dividing each side by the same non-zero number

In our example \(-y = 8\), we used multiplication to manipulate the equation. By multiplying both sides by \(-1\), we effectively reversed the sign of \(y\), helping to 'flip' the equation with minimal effort. Through these manipulations, we aim to simplify the equation step by step, making it easier to isolate and solve for the variable.

Techniques for Isolating Variables

Isolating the variable is a crucial step in solving \(y\) in the equation. Our mission is to get the variable by itself on one side of the equation. This allows us to clearly see the value of the variable. There's a simple strategy to remember:

• Do the opposite operation to cancel out unwanted terms.

In \(-y = 8\), \(y\) is being multiplied by \(-1\). By multiplying by \(-1\) again, we cancel this operation, isolating \(y\). It's like balancing a scale—whatever you do to one side, do to the other. This ensures fairness in the equation and helps keep the relationship between both sides intact. After isolating, we found \(y = -8\). Easy and effective!

Why Checking Solutions is Important

Once you've solved for a variable, it's essential to check your solution. This verifies the accuracy of your work and catches any possible mistakes. Why is it so important? Because it ensures:

• The solution satisfies the original equation
• You've applied correct algebraic processes

To check our solution of \(y = -8\), we substitute it back into the original equation \(-y = 8\). This gives us \(-(-8) = 8\), which simplifies to \(8 = 8\). This correctly confirms our earlier calculation. Checking is a quick step but offers invaluable peace of mind that your derived solution is indeed correct. Always remember, trust but verify.