Question
Andy wrote the equation of a line that has a slope of Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction. and passes through the point (3, –2) in function notation.
Step 1: y – (–2) = Step 1: y minus left-parenthesis negative 2 right-parenthesis equals StartFraction 3 Over 4 left-parenthesis x minus 3 right-parenthesis. (x – 3)
Step 2: y + 2 = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x –
Step 3: y + 2 + 2 = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x – + 2
Step 4: y = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x –
Step 5: f(x) = Step 2: y plus 2 equals StartFraction 3 Over 4 EndFraction x minus StartFraction 9 Over 4 EndFraction.x –
Analyze each step to identify if Andy made an error.
Yes, he made an error in Step 1. He switched the x and y values.
Yes, he made an error in Step 2. He did not distribute StartFraction 3 Over 4 EndFraction properly.
Yes, he made an error in Step 3. He should have subtracted 2 from both sides.
No, his work is correct.
Answers
GPT-4o mini
Let's analyze each step provided by Andy to identify any possible errors.
### Step 1:
**Equation:**
\[ y - (-2) = \frac{3}{4}(x - 3) \]
**Analysis:**
This step is correct. He is using the point-slope form of a line equation, and since the point is (3, -2), writing \( y - (-2) \) is appropriate.
### Step 2:
**Equation:**
\[ y + 2 = \frac{3}{4}x - \frac{9}{4} \]
**Analysis:**
In this step, Andy correctly applied the point-slope form to expand out the equation. The right-hand side simplifies correctly from \( \frac{3}{4}(x - 3) \) to \( \frac{3}{4}x - \frac{9}{4} \). So, this step is correct.
### Step 3:
**Equation:**
**Step 3 is not explicitly written here perhaps, but he seems to have written something related to y + 2**, likely checking the operation done in step 2.
**Expected Correction:**
If he added 2 to both sides here for clarification, he should be reinforcing the previous step rather than changing it.
### Step 4:
**Equation:**
This seems to consistently follow the transformation from the previous step to bring \( y \) to one side.
**Analysis:**
From \( y + 2 = \frac{3}{4}x - \frac{9}{4} \), if he wants to isolate \( y \), he subtracts 2. The transition to step 4 appears correct.
### Step 5:
**Equation:**
It appears he is expressing \( y \) as a function of \( x \).
**Final Analysis:**
Since he equated everything as proposed earlier, this step appears correct.
### Conclusion:
No errors have been made in the identification of points or distribution, therefore:
**No, his work is correct.**
### Step 1:
**Equation:**
\[ y - (-2) = \frac{3}{4}(x - 3) \]
**Analysis:**
This step is correct. He is using the point-slope form of a line equation, and since the point is (3, -2), writing \( y - (-2) \) is appropriate.
### Step 2:
**Equation:**
\[ y + 2 = \frac{3}{4}x - \frac{9}{4} \]
**Analysis:**
In this step, Andy correctly applied the point-slope form to expand out the equation. The right-hand side simplifies correctly from \( \frac{3}{4}(x - 3) \) to \( \frac{3}{4}x - \frac{9}{4} \). So, this step is correct.
### Step 3:
**Equation:**
**Step 3 is not explicitly written here perhaps, but he seems to have written something related to y + 2**, likely checking the operation done in step 2.
**Expected Correction:**
If he added 2 to both sides here for clarification, he should be reinforcing the previous step rather than changing it.
### Step 4:
**Equation:**
This seems to consistently follow the transformation from the previous step to bring \( y \) to one side.
**Analysis:**
From \( y + 2 = \frac{3}{4}x - \frac{9}{4} \), if he wants to isolate \( y \), he subtracts 2. The transition to step 4 appears correct.
### Step 5:
**Equation:**
It appears he is expressing \( y \) as a function of \( x \).
**Final Analysis:**
Since he equated everything as proposed earlier, this step appears correct.
### Conclusion:
No errors have been made in the identification of points or distribution, therefore:
**No, his work is correct.**