What do the following two equations represent?
- $y-8=\frac{1}{2}(x+5)$
- $y-8=-\frac{1}{2}(x+5)$

Choose 1 answer:
(A) The same line
(B) Distinct parallel lines
(C) Perpendicular lines
(D) Intersecting, but not perpendicular lines

Question

What do the following two equations represent?
- $y-8=\frac{1}{2}(x+5)$
- $y-8=-\frac{1}{2}(x+5)$

Choose 1 answer:
(A) The same line
(B) Distinct parallel lines
(C) Perpendicular lines
(D) Intersecting, but not perpendicular lines

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To understand what the two equations represent, we analyze their slopes and y-intercepts. ‖ ‖ step 2 ⋮ The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. ‖ ‖ step 3 ⋮ The first equation $y - 8 = \frac{1}{2}(x + 5)$ can be rewritten as $y = \frac{1}{2}x + \frac{5}{2} + 8$. ‖ ‖ step 4 ⋮ The second equation $y - 8 = -\frac{1}{2}(x + 5)$ can be rewritten as $y = -\frac{1}{2}x - \frac{5}{2} + 8$. ‖ ‖ step 5 ⋮ Both equations have the same y-intercept, $8 + \frac{5}{2} = \frac{21}{2}$, but opposite slopes of $\frac{1}{2}$ and $-\frac{1}{2}$. ‖ ‖ step 6 ⋮ Since the slopes are non-zero and negative reciprocals of each other, the lines are perpendicular. ‖ ∻[1] Answer∻ ***C*** ∻Key Concept∻ ⚹Perpendicular Lines⚹ ∻Explanation∻ ⚹Two lines are perpendicular if their slopes are negative reciprocals of each other. In this case, the slopes are $\frac{1}{2}$ and $-\frac{1}{2}$, which are negative reciprocals.⚹◊What are the slopes of the lines passing through the points $(-8,1)$ and $(4,4)$, and $(-9,-7)$ and $(9,-3)$?⍭ Generate me a similar question◊

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