Select the correct answer.
Find the mistake made in the steps to solve the equation below.
$\begin{aligned}
6 x-1 & = -2 x+9 \\
8 x-1 & = 9 \\
8 x & = 10 \\
x & =\frac{8}{10} \\
x & =\frac{4}{5}
\end{aligned}$
1. Addition property of equality
2. Addition property of equality
3. Division property of equality
4. Simplification
A. Step 2 is incorrect and should be $8 x=8$.
B. The justification for step 3 is incorrect and should be the multiplication property of equality.
C. The justification for step 2 is incorrect and should be the subtraction property of equality.
D. Step 3 is incorrect and should be $x=\frac{10}{8}$.
Answer (2)
Add 2 x to both sides: 8 x − 1 = 9 .
Add 1 to both sides: 8 x = 10 .
Divide both sides by 8 : x = 8 10 .
Simplify: x = 4 5 . The mistake is in step 3, where it incorrectly states x = 10 8 . The correct answer is D .
Explanation
Analyzing the Problem We are given the equation 6 x − 1 = − 2 x + 9 and a step-by-step solution that contains an error. Our goal is to identify the step where the mistake was made. We will carefully examine each step to pinpoint the error in the algebraic manipulations.
Identifying the Error Let's analyze each step:
Step 1: 6 x − 1 = − 2 x + 9 becomes 8 x − 1 = 9 . This is achieved by adding 2 x to both sides of the equation: 6 x + 2 x − 1 = − 2 x + 2 x + 9 , which simplifies to 8 x − 1 = 9 . This step is correct.
Step 2: 8 x − 1 = 9 becomes 8 x = 10 . This is achieved by adding 1 to both sides of the equation: 8 x − 1 + 1 = 9 + 1 , which simplifies to 8 x = 10 . This step is also correct.
Step 3: 8 x = 10 becomes x = 10 8 . This step is incorrect. To isolate x , we should divide both sides by 8 , resulting in x = 8 10 , not x = 10 8 .
Step 4: x = 10 8 becomes x = 5 4 . This simplification is correct based on the incorrect result from Step 3, but since Step 3 is wrong, this step is also based on a mistake.
Conclusion The mistake is in Step 3. The equation 8 x = 10 should lead to x = 8 10 after dividing both sides by 8. The solution incorrectly states x = 10 8 . Therefore, the correct answer is D.
Examples
When solving linear equations, it's crucial to perform the same operation on both sides to maintain equality. For instance, if you're balancing a checkbook and need to solve for an unknown expense, correctly applying algebraic operations ensures your balance is accurate. Similarly, in physics, when calculating forces or velocities, accurate algebraic manipulation is essential for precise results. This exercise reinforces the importance of careful and correct algebraic steps in various real-world applications.
The mistake in solving the equation 6 x − 1 = − 2 x + 9 occurs in Step 3, where it incorrectly states that x = 10 8 instead of the correct x = 8 10 . Thus, the chosen option is D. The correct process leads to x = 4 5 after proper simplification.
;
