Let's analyze Helena's work step by step to identify any errors.
Step 1: Finding the slope (m)
Given points: (5, 1) and (3, 5)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points:
\[ m = \frac{5 - 1}{3 - 5} = \frac{4}{-2} = -2 \]
This calculation is correct.
Step 2: Writing the point-slope form equation
She uses the point (3, 5) (which corresponds to \( (x_1, y_1) \) from her calculation):
The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -2 \) and the point (3, 5):
\[ y - 5 = -2(x - 3) \]
Here, Helena used (3, 5) correctly but instead wrote it as \( y - 1 = -2(x - 3) \), using the wrong y-coordinate. This is an error.
Step 3: Simplifying the equation
Helena continues with:
From the above, if we were to correct this:
\[ y - 5 = -2(x - 3) \]
Expanding gives:
\[ y - 5 = -2x + 6 \]
Adding 5 to both sides:
\[ y = -2x + 11 \]
Helena instead got:
\[ y - 1 + 1 = -2x + 6 + 1 \] which is incorrect.
To summarize:
- First Step: No error found; she correctly calculated the slope.
- Second Step: Helena made an error by using the wrong y-coordinate in the point-slope form.
- Third Step: Therefore, when she simplified it, she arrived at the wrong equation because of the earlier mistake.
So the correct answer is: In step 2, she didn’t use an x and y from the same coordinate pair.
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