A line in the xy-plane passes through the point (4,-1) and is perpendicular to the line with equation y=x+5. Which of the following is an equation of the line ? Choose 1 answer: (A) y=x+3 (B) y=x-3 (C) y=-x+3 (D) y=-x-3

Question

A line in the  xy-plane passes through the point  (4,-1) and is perpendicular to the line with equation  y=x+5. Which of the following is an equation of the line ? Choose 1 answer: (A)  y=x+3 (B)  y=x-3 (C)  y=-x+3 (D)  y=-x-3

Bytelearn · Accepted Answer

Finding the slope of the given line: The slope of the given line y = x + 5 is 1, since it is in the form y = mx + b where m is the slope. We need to find the slope of the line that is perpendicular to this line.Determining the slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Since the original slope is 1, the negative reciprocal is -1.Using the point-slope form to find the equation: Now we have the slope of the perpendicular line, which is -1. We can use the point-slope form of the equation of a line to find the equation of our line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point the line passes through.Simplifying the equation: Plugging in the slope -1 and the point (4, -1) into the point-slope form, we get y - (-1) = -1(x - 4).Isolating y in the equation: Simplifying the equation, we get y + 1 = -1(x - 4). Distributing the -1, we get y + 1 = -x + 4.Matching the equation with the answer choices: Subtracting 1 from both sides to get y by itself, we get y = -x + 4 - 1, which simplifies to y = -x + 3.Comparing the equation y = -x + 3 with the answer choices, we find that it matches with option (C) y = -x + 3.

A line in the $x y$ -plane passes through the point $(4,-1)$ and is perpendicular to the line with equation $y=x+5$ . Which of the following is an equation of the line ? $\newline$ Choose $1$ answer: $\newline$ (A) $y=x+3$ $\newline$ (B) $y=x-3$ $\newline$ (C) $y=-x+3$ $\newline$ (D) $y=-x-3$

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