Problem 6 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ -8=8+z $$

Step-by-Step Solution

Answer

The solution is \(z = -16\).

Step 1: Isolate the Variable

To solve the equation \(-8 = 8 + z\), start by isolating the variable \(z\). You do this by subtracting 8 from both sides of the equation to get rid of the 8 on the right side. This gives us:\[-8 - 8 = z\]

Step 2: Simplify the Equation

Simplify the left side of the equation \(-8 - 8\) which results in \(-16\). Therefore, the equation simplifies to:\[z = -16\]

Step 3: Verify the Solution

To check if \(z = -16\) is the correct solution, substitute \(z\) with \(-16\) in the original equation and verify if both sides are equal:\[-8 = 8 + (-16)\]Calculate the right side:\[8 + (-16) = -8\]Since both sides are equal \(-8 = -8\), \(z = -16\) is the correct solution.

Key Concepts

Isolating Variables

When solving algebraic equations, isolating the variable is often the first crucial step. In our exercise with the equation \(-8 = 8 + z\), the variable you want to solve for is \(z\). Think of isolating variables as a game of balance. You need to perform the same operation on both sides of the equation to keep it balanced, much like a see-saw where both sides need to have the same weight.

In this case, you'll need to eliminate the number next to our variable. Since there's a \(+8\) on the same side as \(z\), we should subtract \(8\) from both sides. This leaves us with:

• The left side becomes \(-8 - 8\).
• The right side simplifies to just \(z\).

Isolating the variable serves as a way of strategically "untangling" the problem. By doing this, we're one step closer to finding out exactly what \(z\) is.

Simplifying Equations

After isolating the variable, the next logical step is simplifying the equation. This involves cleaning up each side to its simplest form. Process any arithmetic to whittle down the parts into something cleaner. In our specific problem, after subtracting \(8\) from both sides, the equation changed to \(-8 - 8 = z\).

The task is to simplify \(-8 - 8\), which results in \(-16\). This clarifies the equation to:

• Left side stays as \(-16\).
• Right side remains just \(z\).

So we find that \(z = -16\). Simplification makes equations much easier to interpret and solve, clearing your path to the final answer.

Verifying Solutions

After working hard to isolate the variable and simplify the equation, it's good practice to verify your solution. Verification confirms that your answer satisfies the original equation. In our example, after solving for \(z = -16\), we double-check by substituting back into the initial equation \(-8 = 8 + z\):

• Substitute \(z\) with \(-16\).
• Check the calculation: \(8 + (-16) = -8\).

You check if both sides of the equation are equal, which they are in this situation \(-8 = -8\). Verification is like a safety net ensuring you haven't made any mistakes along the way. It helps you build confidence in your problem-solving skills and ensures that your answer is indeed correct.