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Chapter 2: Problem 9

For exercises 1-22, (a) solve. (b) check. $$ -8 k-11=-2 k+61 $$

Short Answer

Expert verified
k = -12. Verified by substituting back into the original equation.

Step by step solution

01

Isolate the variable term on one side

First, move all the terms involving the variable k to one side of the equation. Add 2k to both sides: \[ -8k - 11 + 2k = -2k + 2k + 61 \] This simplifies to: \[ -6k - 11 = 61 \]
02

Solve for the variable

Next, isolate k by first adding 11 to both sides: \[ -6k - 11 + 11 = 61 + 11 \] This simplifies to: \[ -6k = 72 \] Now, divide both sides by -6: \[ k = \frac{72}{-6} \] This simplifies to: \[ k = -12 \]
03

Check the solution

To verify the solution, substitute k = -12 back into the original equation: \[ -8(-12) - 11 = -2(-12) + 61 \] This simplifies to: \[ 96 - 11 = 24 + 61 \] Simplifying both sides: \[ 85 = 85 \] Since both sides are equal, the solution is correct.

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Key Concepts

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

Isolating the Variable
Isolating the variable is a crucial step in solving linear equations. It means getting the variable by itself on one side of the equation. In our exercise, we start with the equation \[ -8k - 11 = -2k + 61 \]. To isolate the variable, we first need to move all the terms involving \( k \) to one side. We can do this by adding \( 2k \) to both sides. This operation cancels out the \( -2k \) on the right side and combines the \( k \)-terms on the left: \[ -8k - 11 + 2k = -2k + 2k + 61 \]. Now, the equation simplifies to \[ -6k - 11 = 61 \], which gets us closer to isolating our variable.
Equation Simplification
The next step is to simplify the equation to isolate \( k \). We want to get rid of the constant terms. In our example, after isolating the terms with \( k \), we simplify further by isolating \( k \): \[ -6k - 11 = 61 \]. To do this, we add \( 11 \) to both sides. This results in \[ -6k - 11 + 11 = 61 + 11 \], which simplifies to \[ -6k = 72 \]. Finally, we divide both sides by \( -6 \) to solve for \( k \): \[ k = \frac{72}{-6} \]. This simplifies to \[ k = -12 \]. At this point, we have effectively simplified the equation and solved for \( k \).
Solution Verification
Verifying your solution ensures that it is correct. To do this, substitute the value of \( k \) back into the original equation. In our problem, the original equation is \[ -8k - 11 = -2k + 61 \] and we found \( k = -12 \). Substitute \( k = -12 \) back into the equation: \[ -8(-12) - 11 = -2(-12) + 61 \]. Simplifying both sides, this becomes \[ 96 - 11 = 24 + 61 \] and further simplifies to \[ 85 = 85 \]. Since both sides are equal, our solution \( k = -12 \) is indeed correct. This verification step is essential as it confirms that our solution holds true in the original equation.

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