What is the equation of the line that contains the points with (x, y) coordinates (-3, 7) and (5, -1)?

Question

What is the equation of the line that contains the points with  (x, y) coordinates (-3, 7) and (5, -1)?

Eva Adamson · Accepted Answer

#x+y=4# If #(x_1,y_1)=(-3,7)# and #(x_2,y_2)=(5,-1)# are points on a straight line, then the slope of that line is #color(white)("XXX")m=(y_2-y_1)/(x_2-x_1)=(-1-7)/(5-(-3))=(-8)/8=-1# Any general point #(x,y)# on this line together with #(x_1,y_1)=(-3,7)#, for example must have this same slope" #color(white)("XXX")(y-y_1)/(x-x_1)=-1# or #color(white)("XXX")(y-7)/(x-(-3))=-1# Simplifying further #color(white)("XXX")y-7=(-1) * (x+3)# #color(white)("XXX")y-7=-x-3# In standard form: #color(white)("XXX")x+y=4#

Eli Assouline · Answer

undefined To find the equation of the line passing through two points $(-3, 7)$ and $(5, -1)$, you can use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Where:
$m$ is the slope of the line, and
$(x_1, y_1)$ is one of the points on the line.

First, calculate the slope using the formula:

$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$

Substitute the coordinates $(-3, 7)$ and $(5, -1)$:

$m = \frac{{-1 - 7}}{{5 - (-3)}} = \frac{{-8}}{{8}} = -1$

Now, choose one of the points, let's say $(-3, 7)$, and substitute into the point-slope form:

$y - 7 = -1(x - (-3))$

Simplify:

$y - 7 = -1(x + 3)$

$y - 7 = -x - 3$

Now, move the $-7$ to the right side:

$y = -x - 3 + 7$

$y = -x + 4$

So, the equation of the line passing through the points $(-3, 7)$ and $(5, -1)$ is $y = -x + 4$.