First, we must determine the slope of the line. The slope can be found by using the formula: m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))m=y2−y1x2−x1
Where mm is the slope and (color(blue)(x_1, y_1)x1,y1) and (color(red)(x_2, y_2)x2,y2) are the two points on the line.
Substituting the values from the points in the problem gives:
m = (color(red)(-5) - color(blue)(1))/(color(red)(-2) - color(blue)(-3)) = (color(red)(-5) - color(blue)(1))/(color(red)(-2) + color(blue)(3)) = -6/1 = -6m=−5−1−2−−3=−5−1−2+3=−61=−6
We can now use the point slope formula to find an equation for the line. The point-slope formula states: (y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))(y−y1)=m(x−x1)
Where color(blue)(m)m is the slope and color(red)(((x_1, y_1))) is a point the line passes through.
Substituting the slope we calculated and the first point from the problem gives:
(y - color(red)(1)) = color(blue)(-6)(x - color(red)(-3))
Solution 1) (y - color(red)(1)) = color(blue)(-6)(x + color(red)(3))
We can also substitute the slope we calculated and the second point from the problem giving:
(y - color(red)(-5)) = color(blue)(-6)(x - color(red)(-2))
Solution 2) (y + color(red)(5)) = color(blue)(-6)(x + color(red)(2))
We can also solve this for y to put the equation in slope intercept form. The slope-intercept form of a linear equation is: y = color(red)(m)x + color(blue)(b)
Where color(red)(m) is the slope and color(blue)(b) is the y-intercept value.
y + color(red)(5) = (color(blue)(-6) xx x) + (color(blue)(-6) xx color(red)(2))
y + color(red)(5) = -6x - 12
y + color(red)(5) - 5 = -6x - 12 - 5
y + 0 = -6x - 17
Solution 3) y = color(red)(-6)x - color(blue)(17)
