Question
Samuel wrote the equation in slope-intercept form using two points of a linear function represented in a table. Analyze the steps Samuel used to write the equation of the line in slope-intercept form.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 7, negative 5, negative 3, negative 1. Column 2 is labeled y with entries 13, 5, negative 3, negative 11. 1: m = StartFraction negative 3 minus 5 Over (negative 3) minus (negative 5) EndFraction = StartFraction negative 8 Over 2 EndFraction = negative 4. 2: 5 = negative 4 (negative 5) + b. 5 = 20 + b. b = negative 15. 3: y = negative 15 x minus 4.
In which step did Samuel make an error?
In step 1, he substituted the x values for y and the y values for x.
In step 2, he made an arithmetic error.
In step 3, he substituted the wrong values for m and b.
Samuel did not make an error.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 7, negative 5, negative 3, negative 1. Column 2 is labeled y with entries 13, 5, negative 3, negative 11. 1: m = StartFraction negative 3 minus 5 Over (negative 3) minus (negative 5) EndFraction = StartFraction negative 8 Over 2 EndFraction = negative 4. 2: 5 = negative 4 (negative 5) + b. 5 = 20 + b. b = negative 15. 3: y = negative 15 x minus 4.
In which step did Samuel make an error?
In step 1, he substituted the x values for y and the y values for x.
In step 2, he made an arithmetic error.
In step 3, he substituted the wrong values for m and b.
Samuel did not make an error.
Answers
GPT-4o mini
Let's analyze Samuel's steps to identify any errors:
1. **Step 1: Calculate the slope (m)**
Samuel used the points (-5, 5) and (-3, -3) to calculate the slope:
\[
m = \frac{-3 - 5}{-3 - (-5)} = \frac{-8}{2} = -4
\]
This calculation is correct. He properly identified the coordinates of the two points and applied the slope formula.
2. **Step 2: Using the slope and one point to find b**
He used the point (-5, 5) to find the y-intercept (b):
\[
5 = -4(-5) + b \\
5 = 20 + b \\
b = 5 - 20 \\
b = -15
\]
Here, Samuel correctly substituted the slope (m = -4) and the coordinates (-5, 5), but he made an arithmetic error in his subtraction from \(5 - 20\), which should give \(b = -15\) correctly (this part is accurate).
3. **Step 3: Write the equation in slope-intercept form**
He wrote the equation of the line as:
\[
y = -4x - 15
\]
It appears there is a mistake in this equation. He initially wrote \(y = -15x - 4\) which doesn't match the correct slope and intercept values.
From this analysis, the correct response is:
**In step 3, he substituted the wrong values for m and b.**
The correct equation should be \(y = -4x - 15\), not the one he wrote.
1. **Step 1: Calculate the slope (m)**
Samuel used the points (-5, 5) and (-3, -3) to calculate the slope:
\[
m = \frac{-3 - 5}{-3 - (-5)} = \frac{-8}{2} = -4
\]
This calculation is correct. He properly identified the coordinates of the two points and applied the slope formula.
2. **Step 2: Using the slope and one point to find b**
He used the point (-5, 5) to find the y-intercept (b):
\[
5 = -4(-5) + b \\
5 = 20 + b \\
b = 5 - 20 \\
b = -15
\]
Here, Samuel correctly substituted the slope (m = -4) and the coordinates (-5, 5), but he made an arithmetic error in his subtraction from \(5 - 20\), which should give \(b = -15\) correctly (this part is accurate).
3. **Step 3: Write the equation in slope-intercept form**
He wrote the equation of the line as:
\[
y = -4x - 15
\]
It appears there is a mistake in this equation. He initially wrote \(y = -15x - 4\) which doesn't match the correct slope and intercept values.
From this analysis, the correct response is:
**In step 3, he substituted the wrong values for m and b.**
The correct equation should be \(y = -4x - 15\), not the one he wrote.