What is an equation of the line that passes through the points (1,-7) and (-1,5)

Answer 1

To find the equation of a line in the plane you need to know the slope of the line and a point that is passes through.

To determine the slope m (when you know two points), determine the change in y divided by the change in x as you move from point 1 to point 2.

The formula:

m = (y₂-y₁)/(x₂-x₁)

m = (5 - (-7)) / -1 - 1) = 12/-2 = -6.

Or,

m = (-7 - 5)/(1 - (-1)) = -12/2 = -6

Now that you know the slope you can use point-slope to determine the equation of the line (using either point):

Use (1, -7)

y - (-7) = -6(x - 1)

y + 7 = -6x + 6

y = -6x - 1 (note the y-intercept is at (0,-1).

Use (-1,5):

y - 5 = -6(x-(-1))

y - 5 = -6(x+1)

y - 5 = -6x - 6

y = -6x - 1

You can also use y = mx + b if you prefer, again using either one of the given points. This allows you to solve for b. Substitute the x-coordinate of the chosen point for x and the y-coordinate for y (along with m = -6).

Use (-1,5):

5 = -6(-1) + b

5 = 6 + b

b = -1

So, y = -6x - 1.

Use (1, - 7)

-7 = -6(1) + b

-7 = -6 + b

b = -1

y = -6x - 1

Check it out here:

desmos.com/calculator/g5bnertoej

Answer 2

Use the slope formula to determine the slope:

( y₂ - y₁ ) / ( x₂ - x₁ )

= [ 5 - ( -7 ) ] / [ ( -1 ) - 1 ]

= 12 / ( -2 )

= -6

Now, use either the point-slope formula or the slope-intercept form to determine the y-intercept:

y = mx + b

(I recommend the slope-intercept form because I think it is a little easier to remember and use.)

( -7 ) = ( -6 ) * 1 + b

-7 = -6 + b

-1 = b

So,

y = -6x -1

You can check your work by using the other given point:

5 = (-6) * (-1) -1

5 = 6 -1 = 5

[Note that there are at least 4 other formats for writing a linear equation. But the problem only asked for one.]