How do you write an equation of a line given (-5, -8) and is perpendicular to 10x – 6y = -11?

Question

Eli Assouline · Accepted Answer

See a solution process below:

The equation in the problem is in Standard Linear form. The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)# Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1 The slope of an equation in standard form is: #m = -color(red)(A)/color(blue)(B)# Given: #color(red)(10)x - color(blue)(6)y = color(green)(-11)# Then the slope is: #m = color(red)(-10)/color(blue)(-6) = 5/3# The formula for the slope of a perpendicular line, #m_p# is: #m_p = -1/m# Substituting gives: #m_p = (-1)/(5/3) = -3/5# Now we can use the point-slope formula to write and equation for the line to solve the problem. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))# Where #color(blue)(m)# is the slope and #(color(red)(x_1, y_1))# is a point the line passes through. Substituting the slope we calculated and the values from the point in the problem gives: #(y - color(red)(-8)) = color(blue)(-3/5)(x - color(red)(-5))# #(y + color(red)(8)) = color(blue)(-3/5)(x + color(red)(5))# If necessary we can solve this equation for #y# to put it in slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)# Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value. #y + color(red)(8) = (color(blue)(-3/5) * x) + (color(blue)(-3/5) * color(red)(5))# #y + color(red)(8) = -3/5x - 3# #y + color(red)(8) - 8 = -3/5x - 3 - 8# #y + 0 = -3/5x - 11# #y = color(red)(-3/5)x - color(blue)(11)#

Eli Assouline · Answer

undefined To write the equation of a line perpendicular to the given line $10x - 6y = -11$ and passing through the point $(-5, -8)$, you need to find the slope of the given line, then determine the negative reciprocal of that slope to get the slope of the perpendicular line. After that, you can use the point-slope form of a linear equation to write the equation of the perpendicular line.