Questions: Math Placement Test - South Microsoft Virtual Events Powe myclasses.southuniversity.edu Math Placement Test 0:28:57 elapsed Question 33 (1 point) Determine if the 2 given lines are perpendicular: 2x-5y=-7, 2y=5x-3 No Yes Question 34 (1 point) Find the slope-intercept equation of the line through (3,2) and (5,10). y=3x-2 y=4x-10 y=4x+10 y=-4x+5 Question 35 (1 point) Find an equation of the line through (1,2) and parallel to y=-3x+4 y=-3x+5 y=3x-5 y=-3x+2 y=-3x+1 Question 36 (1 point) Use < or > to make the statement true: -12.3 ... ...-15.4

Math Placement Test - South Microsoft Virtual Events Powe myclasses.southuniversity.edu Math Placement Test 0:28:57 elapsed

Question 33 (1 point) Determine if the 2 given lines are perpendicular: 2x-5y=-7, 2y=5x-3 No Yes

Question 34 (1 point) Find the slope-intercept equation of the line through (3,2) and (5,10). y=3x-2 y=4x-10 y=4x+10 y=-4x+5

Question 35 (1 point) Find an equation of the line through (1,2) and parallel to y=-3x+4 y=-3x+5 y=3x-5 y=-3x+2 y=-3x+1

Question 36 (1 point) Use < or > to make the statement true: -12.3 ... ...-15.4

Transcript text: Math Placement Test - South Microsoft Virtual Events Powe myclasses.southuniversity.edu Math Placement lest 0:28:57 elapsed Question 33 (1 point) Determine if the 2 given lines are perpendicular: $2 x-5 y=-7,2 y=5 x-3$ No Yes Question 34 (1 point) Find the slope-intercept equation of the line through $(3,2)$ and $(5,10)$. $y=3 x-2$ $y=4 x-10$ $y=4 x+10$ $y=-4 x+5$ Question 35 (1 point) Find an equation of the line through $(1,2)$ and parallel to $y=-3 x+4$ $y=-3 x+5$ $y=3 x-5$ $y=-3 x+2$ $y=-3 x+1$ Question 36 (1 point) Use < or > to make the statement true: $-12.3 \ldots \ldots-15.4$

Solution

Solution Steps

Solution Approach

Question 33: To determine if two lines are perpendicular, we need to check if the product of their slopes is -1. First, convert each line equation to the slope-intercept form $y = mx + b$ to find their slopes.

Question 34: To find the slope-intercept equation of a line through two points, calculate the slope using the formula $(y_2 - y_1) / (x_2 - x_1)$. Then use the point-slope form $y - y_1 = m(x - x_1)$ to find the equation and convert it to slope-intercept form.

Question 35: For a line parallel to a given line, the slope must be the same. Use the slope from the given line and the point-slope form with the new point to find the equation.

Step 1: Determine if the Lines are Perpendicular

The equations of the lines are given as:

$ 2x - 5y = -7 $
$ 2y = 5x - 3 $

Converting these to slope-intercept form $ y = mx + b $:

For the first line: \[ 5y = 2x + 7 \implies y = \frac{2}{5}x + \frac{7}{5} \] Thus, the slope $ m_1 = \frac{2}{5} $.
For the second line: \[ 2y = 5x - 3 \implies y = \frac{5}{2}x - \frac{3}{2} \] Thus, the slope $ m_2 = \frac{5}{2} $.

To check if the lines are perpendicular, we calculate the product of the slopes: \[ m_1 \cdot m_2 = \frac{2}{5} \cdot \frac{5}{2} = 1 \] Since the product is not $-1$, the lines are not perpendicular.

Step 2: Find the Slope-Intercept Equation of the Line Through (3, 2) and (5, 10)

Using the points $ (3, 2) $ and $ (5, 10) $, we calculate the slope: \[ m = \frac{10 - 2}{5 - 3} = \frac{8}{2} = 4 \] Using the point-slope form $ y - y_1 = m(x - x_1) $: \[ y - 2 = 4(x - 3) \implies y - 2 = 4x - 12 \implies y = 4x - 10 \] Thus, the slope-intercept equation is $ y = 4x - 10 $.

Step 3: Find the Equation of the Line Through (1, 2) and Parallel to $ y = -3x + 4 $

The slope of the given line $ y = -3x + 4 $ is $ -3 $. A line parallel to this will have the same slope. Using the point $ (1, 2) $: \[ y - 2 = -3(x - 1) \implies y - 2 = -3x + 3 \implies y = -3x + 5 \] Thus, the equation of the line is $ y = -3x + 5 $.