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Chapter 8: Problem 124

Solve the equation for the variable. $$8 z=7 z-7$$

Short Answer

Expert verified
z = -7

Step by step solution

01

Subtract 7z from both sides

To isolate the variable, start by subtracting 7z from both sides of the equation. This will help to simplify the equation:\[8z - 7z = 7z - 7 - 7z\]which simplifies to:\[z = -7\]
02

Verify the solution

Substitute the value found for the variable back into the original equation to ensure it is correct:\[8(-7) = 7(-7) - 7\]Simplify both sides:\[-56 = -49 - 7\]which simplifies to:\[-56 = -56\]Since both sides are equal, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
To solve a linear equation, our primary goal is to find the value of the variable. This process is called 'isolating the variable'.
In our example, the equation is \(8z = 7z - 7\).
We want to get 'z' by itself on one side of the equation. To do this, perform operations that simplify the equation step by step, ensuring 'z' stands alone.
First, subtract \(7z\) from both sides:
\(8z - 7z = 7z - 7 - 7z\).
By subtracting \(7z\) from both sides, we effectively remove it from the right side, which lets us focus on 'z' more easily. After performing the subtraction, we get a simpler equation:
\(z = -7\).
This isolated 'z', showing that \(z = -7\).
Simplifying Equations
Simplifying equations is a crucial part of solving them. The goal is to reduce the equation to its simplest form so the solution becomes clear.
We began with \(8z = 7z - 7\).
By subtracting \(7z\) from each side:
\[8z - 7z = 7z - 7 - 7z\]
simplifies to
\[z = -7\].
This step removes like terms and greatly simplifies the problem. It's important to balance the equation by performing the same operations on both sides. This keeps our equation equal and helps us confidently solve for the variable.
Verifying Solutions
After finding the solution, it's important to verify it. This ensures that our answer is correct. To verify, substitute the found solution back into the original equation. In our case, we substitute \(z = -7\) back into the equation:
\[8(-7) = 7(-7) - 7\].
Simplify both sides:
\[-56 = -49 - 7\].
This further simplifies to:
\[-56 = -56\].
Since both sides of the equation match, our solution is verified.
Verification confirms that our steps were correct and that \(z = -7\) is indeed the correct solution.

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