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. The justification for step 2 is incorrect and should be the subtraction property of equality. B. The justification for step 3 is incorrect and should be the multiplication property of equality. C. Step 3 is incorrect and should be $x=\frac{10}{8}$. D. Step 2 is incorrect and should be $8 x=8$.
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 should be x = 8 10 .
C
Explanation
Analyzing the Problem Let's analyze the steps to solve the equation 6 x − 1 = − 2 x + 9 and identify any mistakes.
Step 1: Addition Property of Equality Step 1: Add 2 x to both sides of the equation 6 x − 1 = − 2 x + 9 . This gives us 6 x + 2 x − 1 = − 2 x + 2 x + 9 , which simplifies to 8 x − 1 = 9 . This step is correct, and the addition property of equality is correctly applied.
Step 2: Addition Property of Equality Step 2: Add 1 to both sides of the equation 8 x − 1 = 9 . This gives us 8 x − 1 + 1 = 9 + 1 , which simplifies to 8 x = 10 . This step is also correct, and the addition property of equality is correctly applied.
Step 3: Division Property of Equality (Mistake Found) Step 3: Divide both sides of the equation 8 x = 10 by 8 . This should give us x = 8 10 . However, the solution incorrectly states x = 10 8 . This is the mistake.
Step 4: Simplification (Based on the Mistake) Step 4: Simplify the correct result from Step 3, which is x = 8 10 . Simplifying this fraction by dividing both the numerator and the denominator by 2, we get x = 4 5 . The solution incorrectly simplifies 10 8 to 5 4 , which is based on the incorrect result from Step 3.
Conclusion Therefore, the mistake is in Step 3, where the equation should be x = 8 10 instead of x = 10 8 . The correct answer is C.
Examples
When solving for an unknown variable in an equation, it's crucial to perform the same operation on both sides to maintain equality. For instance, if you're calculating the required dose of medicine (x) based on a formula like 8 x = 10 , accurately isolating x (in this case, x = 8 10 = 1.25 ml) ensures the correct dosage is administered. A mistake in this calculation could lead to an under- or overdose, highlighting the importance of precision in mathematical operations in real-life applications.
The mistake in solving the equation occurs in Step 3, where it is incorrectly stated that x = 10 8 . It should actually be x = 8 10 , leading to the correct solution of x = 4 5 . Thus, the chosen answer is C.
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