Question 1 A) Krystina is trying to derive the slope of a line with coordinates (6,−2) and (0,8) using the formula ΔyΔx=y2−y1x2−x1 . Krystina says the slope is −35 because slope =riserun=x2−x1y2−y1=0−68−(−2)=−68+2=−610 . Is Krystina’s calculation for slope correct? If not, what is an appropriate explanation for why she is incorrect? (1 point) Responses Krystina is correct and she simplified the slope correctly. Krystina is correct and she simplified the slope correctly. Krystina is incorrect because she calculated the change in y over the change in x. Krystina is incorrect because she calculated the change in y over the change in x. Krystina is incorrect because she calculated the change in x over the change in y. Krystina is incorrect because she calculated the change in x over the change in y. Krystina is correct because she calculated the change in x over the change in y. Krystina is correct because she calculated the change in x over the change in y. Question 2 A) Marco is trying to determine whether the following two lines are parallel. y=23x+7 y=23x−5 Which statement correctly describes the approach Marco should take based on the best reasoning? Statement #1: Marco says he should verify that the lines have the same slope but different y-intercepts. Statement #2: Marco says he should verify that the lines have the same y-intercept but different slopes. (1 point) Statement #$$ describes the approach Marco should take based on the best reasoning. Question 3 A) Use the table to answer the question. Line Equation Line A y=2x−3 Line B y=2x+5 Line C y=−12x−2 Line D y=12x+2 Given the linear equations in the table, which of the following statements is true? (1 point) Responses Line B is perpendicular to line D . Line upper B is perpendicular to line upper D . Line A is perpendicular to line B . Line upper A is perpendicular to line upper B . Line A is perpendicular to line C . Line upper A is perpendicular to line upper C . Line C is perp

Question

Question 1 A) Krystina is trying to derive the slope of a line with coordinates (6,−2)  and (0,8)  using the formula ΔyΔx=y2−y1x2−x1 .  Krystina says the slope is −35  because slope =riserun=x2−x1y2−y1=0−68−(−2)=−68+2=−610 .  Is Krystina’s calculation for slope correct? If not, what is an appropriate explanation for why she is incorrect?  (1 point) Responses  Krystina is correct and she simplified the slope correctly. Krystina is correct and she simplified the slope correctly.  Krystina is incorrect because she calculated the change in y over the change in x. Krystina is incorrect because she calculated the change in y over the change in x.  Krystina is incorrect because she calculated the change in x over the change in y. Krystina is incorrect because she calculated the change in x over the change in y.  Krystina is correct because she calculated the change in x over the change in y.  Krystina is correct because she calculated the change in x over the change in y.  Question 2 A) Marco is trying to determine whether the following two lines are parallel.  y=23x+7  y=23x−5  Which statement correctly describes the approach Marco should take based on the best reasoning?  Statement #1: Marco says he should verify that the lines have the same slope but different y-intercepts.  Statement #2: Marco says he should verify that the lines have the same y-intercept but different slopes.  (1 point) Statement #$$ describes the approach Marco should take based on the best reasoning. Question 3 A) Use the table to answer the question.  Line Equation Line A y=2x−3 Line B y=2x+5 Line C y=−12x−2 Line D y=12x+2 Given the linear equations in the table, which of the following statements is true?  (1 point) Responses  Line B  is perpendicular to line D . Line  upper B is perpendicular to line upper D .  Line A  is perpendicular to line B . Line  upper A is perpendicular to line upper B .  Line A  is perpendicular to line C . Line  upper A is perpendicular to line upper C .  Line C  is perp

GPT-4o mini · Answer

### 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: 1. $ y = \frac{2}{3}x + 7 $ (slope = $\frac{2}{3}$) 2. $ 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: 1. **Line A: $ y = 2x - 3 $** (slope = 2) 2. **Line B: $ y = 2x + 5 $** (slope = 2) 3. **Line C: $ y = -\frac{1}{2}x - 2 $** (slope = -\frac{1}{2}) 4. **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.

Answer

Question 1 A) Krystina is trying to derive the slope of a line with coordinates (6,−2)  and (0,8)  using the formula ΔyΔx=y2−y1x2−x1 . Krystina says the slope is −35  because slope =riserun=x2−x1y2−y1=0−68−(−2)=−68+2=−610 . Is Krystina’s calculation for slope correct? If not, what is an appropriate explanation for why she is incorrect? (1 point) Responses Krystina is correct and she simplified the slope correctly. Krystina is correct and she simplified the slope correctly. Krystina is incorrect because she calculated the change in y over the change in x. Krystina is incorrect because she calculated the change in y over the change in x. Krystina is incorrect because she calculated the change in x over the change in y. Krystina is incorrect because she calculated the change in x over the change in y. Krystina is correct because she calculated the change in x over the change in y.  Krystina is correct because she calculated the change in x over the change in y.  Question 2 A) Marco is trying to determine whether the following two lines are parallel. y=23x+7 y=23x−5 Which statement correctly describes the approach Marco should take based on the best reasoning? Statement #1: Marco says he should verify that the lines have the same slope but different y-intercepts. Statement #2: Marco says he should verify that the lines have the same y-intercept but different slopes. (1 point) Statement #$$ describes the approach Marco should take based on the best reasoning. Question 3 A) Use the table to answer the question. Line Equation Line A y=2x−3 Line B y=2x+5 Line C y=−12x−2 Line D y=12x+2 Given the linear equations in the table, which of the following statements is true? (1 point) Responses Line B is perpendicular to line D . Line  upper B is perpendicular to line upper D . Line A is perpendicular to line B . Line  upper A is perpendicular to line upper B . Line A is perpendicular to line C . Line  upper A is perpendicular to line upper C . Line C is perp

GPT-4o mini · Answer

### Question 1 **Krystina's Calculation and Explanation:** To find the slope using the coordinates $(x_1, y_1) = (6, -2)$ and $(x_2, y_2) = (0, 8)$, we use the slope formula: $$ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} $$ Calculating: - $y_2 - y_1 = 8 - (-2) = 8 + 2 = 10$ - $x_2 - x_1 = 0 - 6 = -6$ Thus, the slope is: $$ \text{slope} = \frac{10}{-6} = -\frac{5}{3} $$ **Conclusion on Krystina's Calculation:** Krystina's assertion of a slope of $-35$ is incorrect. She has misunderstood the slope formula and made errors in substitution and simplification. ### 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, we need to check if they have the same slope and different y-intercepts. Given the lines: 1. $ y = \frac{2}{3}x + 7 $ (slope = $\frac{2}{3}$) 2. $ y = \frac{2}{3}x - 5 $ (slope = $\frac{2}{3}$) Both lines have the same slope of $\frac{2}{3}$ but different y-intercepts (7 and -5), meaning they are indeed parallel. ### Correct Statement: **Statement #1: Marco says he should verify that the lines have the same slope but different y-intercepts.** --- ### Question 3 **Analyzing the Slopes:** - **Line : $ 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 Perpendicularity:** Two lines are perpendicular if the product of their slopes is $-1$. - The slopes of Line A and Line B are both $2$, so they are parallel. - The slopes of Line C (-\frac{1}{2}) and Line D (\frac{1}{2}) yield a product of: $$ -\frac{1}{2} \times \frac{1}{2} = -\frac{1}{4} $$ This indicates that neither pairs (A & B, C & D) yield perpendicular lines, according to the traditional slopes. ### Correct Statement: None of the provided statements about the lines in the table correctly represent the relationships among them with regard to perpendicularity. However, none suggest Line A is perpendicular to line C as well as the other pairings. Assuming you have other test selections to consider, let's clarify each provided statement: - **Line B is perpendicular to line D**: Incorrect (same slope) - **Line A is perpendicular to line B**: Incorrect (same slope) - **Line A is perpendicular to line C**: Incorrect (not $-1$) - **Line C is perpendicular to other Lines based on slope relations**: Not correct. You can summarize that no pair is perpendicular based on the given responses. Adjust selections if needed based on clarifying statements. If meant to conclude clarity, please use appropriate context as needed.