Question

find the slope of the line passing through the pairs of points and describe the line as rising, falling, horizontal or ···(-1, 2), (-1, -5)(-1, 0), (3, -5)determine whether the graphs of each pair of equations are parallel, perpendicular or neither.$y = \frac{5}{6}x - 6LXB0y = -\frac{1}{3}x + 2LXB1y = -4x + 1LXB2y = 4LXB3y + 2x = 4; (-1, 2)$

Explanation:

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Part 1: Slope Calculation

Step1: Slope formula for points

Slope $m=\frac{y_2-y_1}{x_2-x_1}$

Step2: Calculate for (-1,2), (-1,-5)

$m=\frac{-5-2}{-1-(-1)}=\frac{-7}{0}$ (undefined, vertical line)

Step3: Calculate for (-1,0), (3,-5)

$m=\frac{-5-0}{3-(-1)}=\frac{-5}{4}$ (negative = falling)
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Part 2: Parallel/Perpendicular Check

Step1: Convert to slope-intercept form

Slope-intercept: $y=mx+b$, $m$=slope

Step2: Analyze Pair 1

$x+5y=4$ → $y=-\frac{1}{5}x+\frac{4}{5}$. $m_1=\frac{5}{6}$, $m_2=-\frac{1}{5}$ (no parallel/perpendicular relation)

Step3: Analyze Pair 2

$m_1=-\frac{1}{3}$, $m_2=3$. $m_1 \times m_2 = -1$ (perpendicular)

Step4: Analyze Pair 3

$4y=x+3$ → $y=\frac{1}{4}x+\frac{3}{4}$. $m_1=-4$, $m_2=\frac{1}{4}$. $m_1 \times m_2 = -1$ (perpendicular)

Step5: Analyze Pair 4

$4y=6$ → $y=\frac{3}{2}$. $m_1=0$, $m_2=0$ (parallel, horizontal lines)
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Part 3: Parallel Line Equations

Step1: Parallel lines have equal slope

Use point-slope: $y-y_1=m(x-x_1)$

Step2: For $y=\frac{1}{2}x+5$; (4,-5)

$m=\frac{1}{2}$. $y-(-5)=\frac{1}{2}(x-4)$ → $y+5=\frac{1}{2}x-2$ → $y=\frac{1}{2}x-7$

Step3: For $y+2x=4$; (-1,2)

Rewrite to $y=-2x+4$, $m=-2$. $y-2=-2(x-(-1))$ → $y-2=-2x-2$ → $y=-2x$

Answer:

1. For points (-1, 2), (-1, -5): Undefined slope, vertical (undefined direction)

For points (-1, 0), (3, -5): Slope = $\frac{-5}{4}$, falling

1. Pair 1: $y=\frac{5}{6}x - 6$ and $x+5y=4$: Neither

Pair 2: $y=-\frac{1}{3}x + 2$ and $y=3x - 5$: Perpendicular
Pair 3: $y=-4x + 1$ and $4y=x + 3$: Perpendicular
Pair 4: $y=4$ and $4y=6$: Parallel

1. For $y=\frac{1}{2}x + 5$; (4, -5): $y=\frac{1}{2}x - 7$

For $y + 2x = 4$; (-1, 2): $y=-2x$