First, we need to determine the slope of the line passing through the two points from the problem. 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)(11) - color(blue)(-3))/(color(red)(5) - color(blue)(2)) = (color(red)(11) + color(blue)(3))/(color(red)(5) - color(blue)(2)) = 14/3m=11−−35−2=11+35−2=143
Now, we can 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)(-3)) = color(blue)(14/3)(x - color(red)(2))
color(green)(Solution 1)) (y + color(red)(3)) = color(blue)(14/3)(x - color(red)(2))
We can also substitute the slope we calculated and the second point from the problem giving:
color(green)(Solution 2)) (y - color(red)(11)) = color(blue)(14/3)(x - color(red)(5))
We can also solve one of these equations for y to put it into the 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)(3) = (color(blue)(14/3) xx x) - (color(blue)(14/3) xx color(red)(2))
y + color(red)(3) = 14/3x - 28/3
y + color(red)(3) - 3 = 14/3x - 28/3 - 3
y + 0 = 14/3x - 28/3 - (3 xx 3/3)
y = 14/3x - 28/3 - 9/3
color(green)(Solution 3)) y = color(red)(14/3)x - color(blue)(37/3)
