Question

What is an equation of the line that passes through the point left bracket, 8, comma, minus, 2, right bracket(8,−2) and is perpendicular to the line 4, x, minus, 3, y, equals, 274x−3y=27?

Answer 1

To find the equation of a line perpendicular to 4x - 3y = 27 and passing through (8, -2), determine the slope of the original line, calculate its negative reciprocal for the perpendicular slope, and use the point-slope form to form the equation, simplifying to y = -3/4x + 4.

Step-by-step explanation:

To find an equation of a line that is perpendicular to a given line and passes through a specific point, we first need to determine the slope of the given line. The given line is 4x - 3y = 27, which can be written in slope-intercept form as y = mx + b to determine the slope. Re-arranging the equation, we get 3y = 4x - 27, or y = (4/3)x - 9. Thus, the slope of the given line is 4/3.

Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the line we are looking for is the negative reciprocal of 4/3, which is -3/4.

Using the point-slope form of the equation y - y₁ = m(x - x₁) and the slope -3/4, we plug in the point (8, -2) to get y - (-2) = -3/4(x - 8). Simplifying, y + 2 = -3/4x + 6. Finally, we write the equation in slope-intercept form, y = -3/4x + 4.