Question

find the mistake made in the steps to solve the equation below.
\\(6x - 1 = -2x + 9\\)
\\(8x - 1 = 9\\) \\(\quad\quad\quad\\) 1. addition property of equality
\\(\quad 8x = 10\\) \\(\quad\quad\quad\\) 2. addition property of equality
\\(\quad\quad x = \frac{8}{10}\\) \\(\quad\quad\\) 3. division property of equality
\\(\quad\quad x = \frac{4}{5}\\) \\(\quad\quad\\) 4. simplification
\\(\bigcirc\\) a. the justification for step 2 is incorrect and should be the subtraction property of equality.
\\(\bigcirc\\) b. step 3 is incorrect and should be \\(x = \frac{10}{8}\\).
\\(\bigcirc\\) c. the justification for step 3 is incorrect and should be the multiplication property of equality.
\\(\bigcirc\\) d. step 2 is incorrect and should be \\(8x = 8\\).

Explanation:

Step1: Analyze Step 2

From \(8x - 1 = 9\), we use the addition property of equality (add 1 to both sides). So \(8x - 1 + 1 = 9 + 1\), which simplifies to \(8x = 10\)? Wait, no—\(9 + 1 = 10\)? Wait, no, \(9 + 1\) is 10? Wait, no, original equation step: from \(8x - 1 = 9\), adding 1 to both sides: \(8x - 1 + 1 = 9 + 1\) → \(8x = 10\)? Wait, no, wait the mistake is in Step 2's calculation? Wait, no, let's check each option.

Option D: Step 2 is incorrect. Let's recalculate Step 2. From \(8x - 1 = 9\), add 1 to both sides: \(8x - 1 + 1 = 9 + 1\) → \(8x = 10\)? Wait, no, \(9 + 1\) is 10? Wait, no, wait the correct calculation: \(8x - 1 = 9\), add 1 to both sides: \(8x = 9 + 1 = 10\)? Wait, but option D says Step 2 should be \(8x = 8\). Wait, that would mean \(9 + 1 = 8\), which is wrong. Wait, maybe I made a mistake. Wait, original equation: \(6x - 1 = -2x + 9\), Step 1: add \(2x\) to both sides: \(8x - 1 = 9\) (correct). Step 2: add 1 to both sides: \(8x - 1 + 1 = 9 + 1\) → \(8x = 10\) (so Step 2's calculation is \(8x = 10\), but option D says Step 2 should be \(8x = 8\). Wait, that would be if we added -1, but no. Wait, maybe the mistake is in Step 2's calculation. Wait, let's check the options again.

Option D: Step 2 is incorrect and should be \(8x = 8\). Let's see: from \(8x - 1 = 9\), add 1 to both sides: \(8x = 9 + 1 = 10\). But if we made a mistake here, maybe the original equation's Step 2 is wrong. Wait, no—wait, maybe the error is in Step 2's calculation. Wait, \(9 + 1 = 10\), but if we did \(9 - 1 = 8\), that would be wrong. Wait, the addition property of equality: adding the same number to both sides. So from \(8x - 1 = 9\), add 1 to both sides: \(8x = 10\) (correct). But option D says Step 2 should be \(8x = 8\), which would be adding -1, but that's subtraction. Wait, no, the addition property allows adding a positive or negative. Wait, maybe the mistake is in Step 2's calculation. Wait, let's check the options:

Option A: Justification for Step 2 is wrong (subtraction property). But Step 2 is adding 1, so addition property is correct. So A is wrong.

Option B: Step 3: \(x = 8/10\) (from \(8x = 10\), divide both sides by 8: \(x = 10/8\), not \(8/10\). Wait, Step 3: \(x = 8/10\) is wrong. Wait, the division property: divide both sides by 8. So \(x = 10/8\), not \(8/10\). But option B says Step 3 should be \(x = 10/8\). Wait, but the options:

Wait, let's re-express:

Step 1: \(6x - 1 = -2x + 9\) → add \(2x\) to both sides: \(8x - 1 = 9\) (correct, Step 1: addition property).

Step 2: add 1 to both sides: \(8x - 1 + 1 = 9 + 1\) → \(8x = 10\) (Step 2: addition property, correct justification, but calculation? Wait, no, \(9 + 1 = 10\), so Step 2's calculation is \(8x = 10\), but option D says Step 2 should be \(8x = 8\). Wait, that would mean \(9 + 1 = 8\), which is wrong. Wait, maybe I messed up. Wait, let's check the options again.

Option D: Step 2 is incorrect. Let's see: if Step 2 is \(8x = 8\), that would mean \(9 + 1 = 8\), which is wrong. So maybe the mistake is in Step 2's calculation. Wait, no—wait, original equation: \(6x - 1 = -2x + 9\). Step 1: \(8x - 1 = 9\) (correct). Step 2: add 1 to both sides: \(8x = 10\) (so Step 2 is correct? But option D says Step 2 is incorrect. Wait, maybe the error is in Step 2's calculation. Wait, no, \(9 + 1 = 10\), so \(8x = 10\) is correct. Then Step 3: divide both sides by 8: \(x = 10/8 = 5/4\), but in the steps, Step 3 is \(x = 8/10\), which is wrong (division property: divide both sides by 8, so \(x = 10/8\), not \(8/10\)). But option B says Step 3 should be \(x = 10/8\). Wait, but let's check the options again.

Wait, the options:

A: Justification for Step 2 is wrong (subtraction property). But Step 2 is addition (adding 1), so justification is correct. So A is wrong.

B: Step 3 is incorrect (should be \(x = 10/8\)). Step 3: from \(8x = 10\), divide both sides by 8: \(x = 10/8\), not \(8/10\). So Step 3's calculation is wrong (swapped numerator and denominator). So B says Step 3 should be \(x = 10/8\), which is correct. Wait, but let's check the options again.

Wait, the original Step 3: \(x = 8/10\) (from \(8x = 10\), dividing both sides by 8: \(x = 10/8\), not \(8/10\)). So Step 3 is incorrect, and should be \(x = 10/8\), which is option B. But wait, let's check the options again.

Wait, the options:

B: Step 3 is incorrect and should be \(x = 10/8\). Yes, because when you divide both sides by 8, \(x = 10/8\), not \(8/10\). So Step 3's calculation is wrong (swapped numerator and denominator). So the mistake is in Step 3.

Wait, but let's check the other options.

Option D: Step 2 is incorrect. Let's see: Step 2 is \(8x = 10\), but option D says it should be \(8x = 8\). That would mean \(9 + 1 = 8\), which is wrong. So D is incorrect.

Option A: Justification for Step 2 is subtraction property. But Step 2 is addition (adding 1), so justification is correct. So A is wrong.

Option C: Justification for Step 3 is multiplication property. But Step 3 is division (dividing by 8), so justification is correct (division property). So C is wrong.

So the correct answer is B? Wait, no, wait the original Step 3: \(x = 8/10\) (from \(8x = 10\), divide both sides by 8: \(x = 10/8\), not \(8/10\)). So Step 3's calculation is wrong, and should be \(x = 10/8\), which is option B.

Wait, but let's recheck:

Step 1: \(6x - 1 = -2x + 9\) → add \(2x\) to both sides: \(8x - 1 = 9\) (correct, addition property).

Step 2: add 1 to both sides: \(8x - 1 + 1 = 9 + 1\) → \(8x = 10\) (correct, addition property).

Step 3: divide both sides by 8: \(x = 10/8\) (not \(8/10\)). So Step 3's calculation is wrong (swapped numerator and denominator), so Step 3 is incorrect and should be \(x = 10/8\), which is option B.

Answer:

B. Step 3 is incorrect and should be \(x = \frac{10}{8}\)