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 3 is incorrect and should be $x=\frac{10}{8}$.
B. The justification for step 2 is incorrect and should be the subtraction property of equality.
C. Step 2 is incorrect and should be $8 x=8$.
D. The justification for step 3 is incorrect and should be the multiplication property of equality
Answer (1)
Add 2 x to both sides of the equation 6 x − 1 = − 2 x + 9 , resulting in 8 x − 1 = 9 .
Add 1 to both sides of 8 x − 1 = 9 , resulting in 8 x = 10 .
Divide both sides of 8 x = 10 by 8, resulting in x = 8 10 .
Simplify the fraction to get the final answer: x = 4 5 .
Explanation
Analyzing the Problem We are given the equation 6 x − 1 = − 2 x + 9 and a step-by-step solution that contains a mistake. Our goal is to identify the step where the mistake was made. Let's analyze each step carefully.
Step 1 Step 1: 6 x − 1 = − 2 x + 9 . This is the given equation.
Step 2 Step 2: 8 x − 1 = 9 . This step is obtained by adding 2 x to both sides of the equation in Step 1: 6 x − 1 + 2 x = − 2 x + 9 + 2 x , which simplifies to 8 x − 1 = 9 . The addition property of equality is correctly applied here.
Step 3 Step 3: 8 x = 10 . This step is obtained by adding 1 to both sides of the equation in Step 2: 8 x − 1 + 1 = 9 + 1 , which simplifies to 8 x = 10 . The addition property of equality is correctly applied here.
Step 4 Step 4: x = 10 8 . This step is obtained by dividing both sides of the equation in Step 3 by 8. However, dividing 8 x = 10 by 8 should result in x = 8 10 , not x = 10 8 . Therefore, this step is incorrect.
Step 5 Step 5: x = 5 4 . This step simplifies the incorrect result from Step 4, so it is also incorrect.
Conclusion The mistake is in Step 4. The correct step should be x = 8 10 , which simplifies to x = 4 5 . Therefore, the correct answer is A.
Examples
Solving linear equations is a fundamental skill in algebra and has numerous real-world applications. For example, suppose you are trying to determine how many hours you need to work at a part-time job to earn enough money to buy a new phone. If the phone costs $200, and you earn $12 per hour, you can set up the equation 12 h = 200 , where h is the number of hours you need to work. Solving for h gives you h = 12 200 = 16.67 hours. This means you need to work approximately 16 hours and 40 minutes to afford the phone. Linear equations help in budgeting, financial planning, and making informed decisions about time and money.
