Find the equation of a line through a given point, parallel to a second line.

Question

Assume that the line passes through the point (2,−4)(2,−4) and is parallel to the line through the points ( −3,2)−3,2) and (−5,−1)(−5,−1).

Answer 1

Parallel means same slope. Slope is the change in the y-component("rise") divided by ("over") the change in the x-component ("run"). The slope of the parallel line is [(-1)-(-2)]/[(-5)-(-3)]= (1)/(-2)= -0.5

Therefore, the slope (m) of the line must be -0.5 for it to be parallel.

 

The general form of a line is y=mx+b. For any given line, the value of slope (m) and the y-intercept (b) will be constant, or in other words, they don't change and they can solved for. x can have any value, and there will be a corresponding y-value for any point on the line. By knowing a point on the line, we can plug-in the (x,y) values to solve for the y-intercept, which is the only remaining unknown constant. -4= (0.5)*(2)+b

b=-5

Therefore, the equation of the line is y=0.5x-5

 

We solved for the values of the constants (m and b) in our final line equation, but the variables (x and y) do not have set values because it represents all possible (x,y) pairs that exist on the line.

 

Answer 2

m = (y₂-y₁)/(x₂-x₁₎

m = (-1-2)/(-5+3) = =3/-2    Parallel lines have the same slope. Now, use the point-slope form for the equation of a line.

y-y₁ = m(x-x₁)

y+4 = (-3/2)(x-2)

y + 4 = (-3/2) x + 3

y = (-3/2)x - 1