Which of the following is an equation of the line in the xy-plane that passes through the point (3,-4) and is perpendicular to the line with equation y=(1)/(4)x-7 ? Choose 1 answer: (A) y=(1)/(4)x+8 (B) y=(1)/(4)x-8 (C) y=-4x+8 (D) y=-4x-8

Question

Which of the following is an equation of the line in the  xy-plane that passes through the point  (3,-4) and is perpendicular to the line with equation  y=(1)/(4)x-7 ? Choose 1 answer: (A)  y=(1)/(4)x+8 (B)  y=(1)/(4)x-8 (C)  y=-4x+8 (D)  y=-4x-8

Bytelearn · Accepted Answer

Finding the slope of the given line: The slope of the given line is the coefficient of x in the equation y=(1)/(4)x-7, which is 1/4.Determining the slope of the perpendicular line: Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of 1/4. The negative reciprocal of 1/4 is -4.Using the point-slope form to find the equation: Now we have the slope of the new line, which is -4, and a point it passes through, which is (3,-4). 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 the slope and the point into the point-slope form, we get y - (-4) = -4(x - 3). Simplifying this, we get y + 4 = -4x + 12.Converting to slope-intercept form: Subtracting 4 from both sides of the equation to get it into slope-intercept form (y = mx + b), we get y = -4x + 12 - 4, which simplifies to y = -4x + 8.Matching the equation with the options: Comparing the equation y = -4x + 8 with the answer choices, we find that it matches with option (C).

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