Line L has equation #2x- 3y=5#. Line M passes through the point (3, -10) and is parallel to line L. How do you determine the equation for line M?

Question

Isaiah Anthony · Accepted Answer

See a solution process below:

Line L 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 #color(red)(2)x - color(blue)(3)y = color(green)(5)# The slope of an equation in standard form is: #m = -color(red)(A)/color(blue)(B)# Substituting the values from the equation into the slope formula gives: #m = color(red)(-2)/color(blue)(-3) = 2/3# Because line M is parallel to line L, Line M will have the same slope. We can now use the point-slope formula to write an equation for Line M. 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)(-10)) = color(blue)(2/3)(x - color(red)(3))# #(y + color(red)(10)) = color(blue)(2/3)(x - color(red)(3))# If necessary for the answer we can transform this equation to the Standard Linear form as follows: #y + color(red)(10) = (color(blue)(2/3) xx x) - (color(blue)(2/3) xx color(red)(3))# #y + color(red)(10) = 2/3x - 2# #color(blue)(-2/3x) + y + color(red)(10) - 10 = color(blue)(-2/3x) + 2/3x - 2 - 10# #-2/3x + y + 0 = 0 - 12# #-2/3x + y = -12# #color(red)(-3)(-2/3x + y) = color(red)(-3) xx -12# #(color(red)(-3) xx -2/3x) + (color(red)(-3) xx y) = 36# #color(red)(2)x - color(blue)(3)y = color(green)(36)#

Kennedy Adams · Answer

undefined To determine the equation for line M, we need to use the fact that parallel lines have the same slope. The slope of line L can be found by rearranging its equation into slope-intercept form (y = mx + b), where m represents the slope.

Given the equation of line L: 2x - 3y = 5

Rearranging to slope-intercept form: 
-3y = -2x + 5
y = (2/3)x - 5/3

The slope of line L is 2/3.

Since line M is parallel to line L, it will also have a slope of 2/3. Now, using the point-slope form of a line (y - y1 = m(x - x1)), where (x1, y1) is a point on the line and m is the slope, we can find the equation for line M.

Given that line M passes through the point (3, -10) and has a slope of 2/3:

y - (-10) = (2/3)(x - 3)
y + 10 = (2/3)(x - 3)

Now, we can simplify and rearrange the equation to slope-intercept form:

y + 10 = (2/3)x - 2
y = (2/3)x - 12

Therefore, the equation for line M is y = (2/3)x - 12.