Question

brians work is shown. where did he make a mistake? choose the first incorrect step in brians work. step 1, because he should not have subtracted 6 to isolate 3k step 2, because 15 - 6 ≠ 9 step 3, because he should not have divided by 3 to isolate k step 4, because \\(\frac{3k}{3} \
eq k\\) brians work \\(3k - 6 = 15\\) step 1 \\(-6\\) \\(-6\\) step 2 \\(3k = 9\\) step 3 \\(\frac{3k}{3} = \frac{9}{3}\\) step 4 \\(k = 3\\)

Explanation:

Step1: Analyze Step 1

To solve \(3k - 6=15\), we need to isolate \(3k\). The correct operation is to add 6 to both sides (since we have \(-6\) on the left). Brian subtracted 6, which is incorrect. But let's check the options. The first incorrect step: Let's check each step's validity.

Step2: Analyze Step 2

In Step 2, Brian has \(3k = 9\). But starting from \(3k-6 = 15\), if we subtract 6 from both sides (as Brian did in Step 1: \(3k-6-6=15 - 6\)? No, wait, Step 1 is \(3k-6-6\)? No, the original equation is \(3k - 6=15\). To isolate \(3k\), we should add 6 to both sides, not subtract 6. But the option about Step 2 says "because \(15 - 6 eq9\)". Wait, if we did subtract 6 (wrong operation), \(15-6 = 9\), but the operation is wrong. Wait, no: the correct operation to solve \(3k - 6=15\) is add 6 to both sides: \(3k-6 + 6=15 + 6\), so \(3k=21\). But Brian subtracted 6: \(3k-6-6=15 - 6\) → \(3k-12 = 9\), which is wrong. But the option for Step 1 says "he should not have subtracted 6 to isolate \(3k\)". Wait, but let's check the options given. The options are:

1. Step 1, because he should not have subtracted 6 to isolate \(3k\)
2. Step 2, because \(15 - 6 eq9\)
3. Step 3, because he should not have divided by 3 to isolate \(k\)
4. Step 4, because \(\frac{3k}{3} eq k\)

Wait, \(15 - 6 = 9\) is true (arithmetic), but the operation in Step 1 is wrong (subtracting 6 instead of adding 6). But the option for Step 2: the reasoning is "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is true. Wait, no, the error is in the operation of Step 1, but let's re - evaluate.

Wait, the original equation is \(3k - 6 = 15\). To solve for \(3k\), we need to get rid of \(-6\) on the left. The inverse operation of subtraction is addition, so we should add 6 to both sides. Brian did Step 1: subtract 6 from both sides: \(3k - 6-6=15 - 6\) → \(3k - 12 = 9\), which is wrong. But the Step 1 option says "he should not have subtracted 6 to isolate \(3k\)". The Step 2 option: "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is correct arithmetic, but the operation leading to Step 2 is wrong (because the operation in Step 1 is wrong). Wait, no, the options are about the first incorrect step. Let's check each:

- Step 1: The goal is to isolate \(3k\) in \(3k - 6=15\). To isolate \(3k\), we need to add 6 to both sides (since we have \(3k - 6\)), so subtracting 6 is incorrect. But the Step 2 option says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is true. So the error in Step 2's reasoning is wrong, but the first incorrect step is Step 1? Wait, no, let's check the steps again.

Wait, Brian's Step 1: \(3k - 6=15\) → subtract 6 from both sides: \(3k - 6-6=15 - 6\) → \(3k - 12 = 9\). Then Step 2: \(3k = 9\) (he must have added 12 to both sides? No, his Step 2 is \(3k = 9\), which is wrong. Wait, maybe I misread his work. Let me re - read Brian's work:

Brian's work:

Step 1: \(3k - 6=15\) → subtract 6 from both sides: \(-6\) \(-6\) (so \(3k - 6-6=15 - 6\))

Step 2: \(3k = 9\) (so he combined \(3k - 6-6\) as \(3k\) and \(15 - 6\) as 9? No, that's incorrect. Wait, no—maybe he thought to isolate \(3k\), he subtracts 6, but that's wrong. The correct way is to add 6. So Step 1 is incorrect because he subtracted 6 (should add 6) to isolate \(3k\). But the Step 2 option says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is correct. So the first incorrect step is Step 1? Wait, no, the options:

Wait, the first option: "Step 1, because he should not have subtracted 6 to isolate \(3k\)"—this is correct. But let's check the other options. The Step 2 option's reasoning is wrong (since \(15 - 6 = 9\) is true), Step 3: dividing by 3 is the correct operation to isolate \(k\) (since \(3k\) divided by 3 is \(k\)), Step 4: \(\frac{3k}{3}=k\) is correct. So the first incorrect step is Step 1? Wait, no, the answer options—wait, maybe I made a mistake. Wait, the equation is \(3k - 6 = 15\). To solve for \(k\), we add 6 to both sides: \(3k=15 + 6=21\), then divide by 3: \(k = 7\). Brian subtracted 6: \(3k-6-6=15 - 6\) → \(3k - 12 = 9\), then Step 2: \(3k=21\)? No, his Step 2 is \(3k = 9\), which is wrong. Wait, his Step 1: he subtracted 6 from both sides: \(3k - 6-6=15 - 6\) → \(3k - 12 = 9\). Then Step 2: \(3k=9 + 12=21\)? No, his Step 2 is \(3k = 9\), which is wrong. But the option for Step 1 says "he should not have subtracted 6 to isolate \(3k\)"—that's the first mistake. But the Step 2 option says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is true. So the correct answer is the first option? Wait, no, the problem says "Choose the first incorrect step". Let's check each step:

Step 1: Operation: subtract 6 from both sides. Correct operation to isolate \(3k\) is add 6. So Step 1 is incorrect.

Step 2: \(3k = 9\). But from Step 1, if we subtract 6 from both sides: \(3k - 6-6=15 - 6\) → \(3k - 12 = 9\), so \(3k=21\), not 9. So Step 2 is also incorrect, but the first incorrect step is Step 1. But the option for Step 2 says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is true. So the error in Step 2 is not the arithmetic of \(15 - 6\), but the result from the wrong operation in Step 1. Wait, maybe the intended answer is the first option: Step 1, because he should not have subtracted 6 to isolate \(3k\). But let's check the options again.

Wait, the other options:

- Step 3: dividing by 3 is correct to isolate \(k\), so that's wrong.

- Step 4: \(\frac{3k}{3}=k\) is correct, so that's wrong.

- Step 2: \(15 - 6 = 9\) is correct, so the reasoning is wrong, but the step's result is wrong because of Step 1's error.

But the first incorrect step is Step 1. However, the option for Step 2 says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is correct. So maybe the intended answer is the first option: Step 1, because he should not have subtracted 6 to isolate \(3k\). Wait, no, let's re - calculate:

Original equation: \(3k - 6 = 15\)

To solve for \(3k\), we need to get rid of \(-6\), so we add 6 to both sides:

\(3k - 6+6 = 15+6\)

\(3k = 21\)

Brian's Step 1: subtract 6 from both sides:

\(3k - 6-6 = 15 - 6\)

\(3k - 12 = 9\)

Then Step 2: \(3k = 9 + 12 = 21\)? No, his Step 2 is \(3k = 9\), which is wrong. So the first incorrect step is Step 1 (subtracting 6 instead of adding 6). So the answer is the first option: Step 1, because he should not have subtracted 6 to isolate \(3k\). Wait, but the option for Step 2 says "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is correct. So the correct answer is the first option. But wait, maybe I made a mistake. Let's check the options again.

Wait, the problem is asking "Where did he make a mistake? Choose the first incorrect step". Let's check each step:

- Step 1: Subtracting 6 from both sides to isolate \(3k\) is incorrect (should add 6). So Step 1 is incorrect.

- Step 2: \(3k = 9\) is incorrect, but it's a result of the incorrect Step 1.

- Step 3: Dividing by 3 is correct.

- Step 4: \(\frac{3k}{3}=k\) is correct.

So the first incorrect step is Step 1. But the option for Step 1 is "Step 1, because he should not have subtracted 6 to isolate \(3k\)".

Wait, but the other option for Step 2: "because \(15 - 6 eq9\)"—but \(15 - 6 = 9\) is true. So the reasoning for Step 2 is wrong, but the step's result is wrong. However, the first incorrect step is Step 1.

But wait, maybe the intended answer is the first option. So the correct answer is the first option: Step 1, because he should not have subtracted 6 to isolate \(3k\).

Answer:

Step 1, because he should not have subtracted 6 to isolate \(3k\)