What is the equation of the line in slope-intercept form that passes through point (1, –2) and is parallel to the line y = 3x – 2? Show all necessary steps.

Question

Question 8 (Essay Worth 2 points)
(04.02 MC)

Line g is shown on the graph.

coordinate plane with line g that passes through the points negative 3 comma 2 and 0 comma 5

What is the equation of the line in slope-intercept form that is perpendicular to line g and passes through the point (4, 2)? Show all necessary steps.

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FDK431.12

GPT-4o mini · Answer

### Question 7 To find the equation of the line in slope-intercept form that is parallel to the line $ y = 3x - 2 $ and passes through the point $ (1, -2) $, follow these steps: 1. **Identify the slope of the given line**: The slope-intercept form of a line is given by the equation: $$ y = mx + b $$ where $ m $ is the slope. From the line $ y = 3x - 2 $, we can see that the slope $ m = 3 $. 2. **Use the same slope for the parallel line**: Since parallel lines have the same slope, the slope of our desired line is also $ m = 3 $. 3. **Use the point-slope form of the equation**: The point-slope form of a line is: $$ y - y_1 = m(x - x_1) $$ Using the point $ (1, -2) $ (where $ x_1 = 1 $ and $ y_1 = -2 $) and the slope $ m = 3 $: $$ y - (-2) = 3(x - 1) $$ 4. **Simplify the equation**: $$ y + 2 = 3(x - 1) $$ $$ y + 2 = 3x - 3 $$ $$ y = 3x - 3 - 2 $$ $$ y = 3x - 5 $$ **Final answer**: The equation of the line in slope-intercept form is $ y = 3x - 5 $. --- ### Question 8 To find the equation of the line in slope-intercept form that is perpendicular to line $ g $ and passes through the point $ (4, 2) $, follow these steps: 1. **Determine the slope of line $ g $**: First, calculate the slope of line $ g $ that passes through the points $ (-3, 2) $ and $ (0, 5) $. The formula for the slope $ m $ is: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Here $ (x_1, y_1) = (-3, 2) $ and $ (x_2, y_2) = (0, 5) $: $$ m = \frac{5 - 2}{0 - (-3)} = \frac{3}{3} = 1 $$ 2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus, the slope of our desired line is: $$ m_{\text{perpendicular}} = -\frac{1}{1} = -1 $$ 3. **Use the point-slope form for the perpendicular line**: Using the point $ (4, 2) $ and the slope $ -1 $: $$ y - 2 = -1(x - 4) $$ 4. **Simplify the equation**: $$ y - 2 = -1(x - 4) $$ $$ y - 2 = -x + 4 $$ $$ y = -x + 4 + 2 $$ $$ y = -x + 6 $$ **Final answer**: The equation of the line in slope-intercept form that is perpendicular to line $ g $ and passes through the point $ (4, 2) $ is $ y = -x + 6 $.