Question 1
Krystina's Calculation:
To find the slope using the formula:
\[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]
Given the coordinates \((x_1, y_1) = (6, -2)\) and \((x_2, y_2) = (0, 8)\):
- \(y_2 - y_1 = 8 - (-2) = 8 + 2 = 10\)
- \(x_2 - x_1 = 0 - 6 = -6\)
Thus, the slope is:
\[ \text{slope} = \frac{10}{-6} = -\frac{5}{3} \]
Conclusion:
Krystina's calculation is incorrect because she made mistakes in substituting the coordinates. Her argument suggested \( -35 \) and her calculation did not follow the correct formula.
Correct Response:
Krystina is incorrect because she calculated the change in y over the change in x.
Question 2
To determine whether two lines are parallel, they must have the same slope but different y-intercepts.
Given the equations:
- \( y = \frac{2}{3}x + 7 \) (slope = \(\frac{2}{3}\))
- \( y = \frac{2}{3}x - 5 \) (slope = \(\frac{2}{3}\))
Since both lines have the same slope \(\frac{2}{3}\) but different y-intercepts (7 and -5), they are parallel.
Correct Statement:
Statement #1: Marco says he should verify that the lines have the same slope but different y-intercepts.
Question 3
To determine which lines are perpendicular, we need to check their slopes. Lines are perpendicular if the product of their slopes is \(-1\).
From the given equations in the table:
- Line A: \( y = 2x - 3 \) (slope = 2)
- Line B: \( y = 2x + 5 \) (slope = 2)
- Line C: \( y = -\frac{1}{2}x - 2 \) (slope = -\frac{1}{2})
- Line D: \( y = \frac{1}{2}x + 2 \) (slope = \frac{1}{2})
Checking for Perpendicularity:
- The slopes of Line A and Line B are the same (2), hence they are parallel.
- The slopes of Line C and Line D are \(-\frac{1}{2}\) and \(\frac{1}{2}\).
- \((-1/2) \cdot (1/2) = -\frac{1}{4}\) (not perpendicular)
Thus, none of the lines A, B, C, or D are perpendicular to each other.
Correct Statement:
After reviewing the slopes, none of the provided statements are true based on the given equations. If considering the context correctly, ensure to carefully derive further conclusions based on possible formatting of the question options.
If you meant:
Line A is perpendicular to line C: Incorrect.
Line B is perpendicular to line D: Incorrect.
Check the provided equations closely for clarification if further review needed.