What is the equation of the line between (3,-13)(3,13) and (-7,1)(7,1)?

2 Answers
May 30, 2018

y = -\frac{7}{5}x - 44/5y=75x445

Explanation:

When you know the coordinates of two points P_1 = (x_1,y_1)P1=(x1,y1) and P_2 = (x_2,y_2)P2=(x2,y2), the line passing through them has equation

\frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1}yy1y2y1=xx1x2x1

Plug your values to get

\frac{y+13}{1+13} = \frac{x-3}{-7-3} \iff \frac{y+13}{14} = \frac{x-3}{-10}y+131+13=x373y+1314=x310

Multiply both sides by 1414:

y+13 = -\frac{7}{5}x + \frac{42}{10}y+13=75x+4210

Subtract 1313 from both sides:

y = -\frac{7}{5}x - 44/5y=75x445

May 30, 2018

Over the top detail given so that you can see where everything comes from.

y=-7/5x-44/5y=75x445

Explanation:

Using the gradient (slope)

Reading left to right on the x-axis.
Set point 1 as P_1->(x_1,y_1)=(-7,1)P1(x1,y1)=(7,1)
Set point 2 as P_2->(x_2,y_2)=(3,-13)P2(x2,y2)=(3,13)

In reading this we 'travel' from x_1x1 to x_2x2 so to determine the difference we have x_2-x_1 and y_2-y_1x2x1andy2y1

color(red)(m)=("change in y")/("change in x") ->(y_2-y_1)/(x_2-x_1)=(-13-1)/(3-(-7)) = color(red)((-14)/(+10)=-7/5) m=change in ychange in xy2y1x2x1=1313(7)=14+10=75

We may choose any of the two: P_1" or "P_2P1 or P2 for the next bit. I choose P_1P1

m=-7/5=(y_2-1)/(x_2-(-7)) =(y_2-1)/(x_2+7)m=75=y21x2(7)=y21x2+7

-7(x_2+7)=5(y_2-1)7(x2+7)=5(y21)

-7x_2-49=5y_2-57x249=5y25

Add 5 to both sides

-7x_2-44=5y_27x244=5y2

Divide both sides by 5

-7/5x_2-44/5=y_275x2445=y2

Now using generic x and yxandy

-7/5x-44/5=y75x445=y

y=-7/5x-44/5y=75x445

Tony B