Unit 4, Algebra 1

Question

Which equation describes a line that passes through (-6, 8) and is parallel to the line described by y = 2x − 4?

Answer 1

The given is a point (-6,8) through which a line passes. And is perpendicular to the line y = 2x-4

The equation for point (-6,8) is,

(y-8)= m1(x+6)...........(i)

and given equation is y = 2x-4............(ii)

Now, from equation (ii).

slope (m2)= 2 { as equation (ii) is in the form of y= mx+c where m is a slope}.

Now, For perpendicular,

m1×m2= -1

m1×2= -1

m1 = \frac{ - 1}{2}m1=2−1​

Therefore, m1 = -1/2.

Putting, the value of m1 in equation (i).

(y-8) = -1/2×(x+6)

2(y-8)= -1(x+6)

2y - 16 = -x -6

x+2y-10 = 0......... is the required equation.

Answer 2

If a line is parallel to another line then it means that they both have the same slope.

The slope of y = 2x - 4 is "2". You can read it right off the equation because the equation is in the form of y = mx + b and slope is "m" so just look for the coefficient in front of the "x" which is "2".

The line you are looking for then has a slope that is "2" and it goes through the point (-6, 8). There are a couple of ways of finding the equation of this line. One is to use the point-slope form of a line. Another is to use the slope-intercept form (y = mx + b). I'll discuss the latter.

Since m = 2, you can substitute "2" in for "m" in y = mx + b to get y = 2x + b.

You are given a point (-6, 8) which is a point (x, y) so x = -6 and y = 8. Plug those into the equation you have (y = 2x + b) to get:

8 = 2(-6) + b

8 = -12 + b

b = 20

So the equation of the line is y = 2x + 20

If you use the point-slope form, which is y - y₁ = m(x - x₁) where "m" is the slope and the point you have is (x₁, y₁) meaning m = 2 and x₁ = -6 and y₁ = 8, you just plug those values into y - y₁ = m(x - x₁):

y - 8 = 2(x - (-6))

y - 8 = 2(x + 6)

Although this version of the equation looks a little different that the first one we developed, they are the same. I'll leave it to you to convince yourself that they are the same.