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))
Where m is the slope and (color(blue)(x_1, y_1)) and (color(red)(x_2, y_2)) are the two points on the line.
Substituting the values from the points in the problem gives:
m = (color(red)(11) - color(blue)(2))/(color(red)(-23) - color(blue)(30)) = 9/-53 = -9/53
We can now use the point-slope formula to find an equation for the line between the two points. The point-slope form of a linear equation is: (y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))
Where (color(blue)(x_1), color(blue)(y_1)) is a point on the line and color(red)(m) is the slope.
Substituting the slope we calculated and the values from the first point in the problem gives:
(y - color(blue)(2)) = color(red)(-9/53)(x - color(blue)(30))
We can also substitute the slope we calculated and the values from the second point in the problem gives:
(y - color(blue)(11)) = color(red)(-9/53)(x - color(blue)(-23))
(y - color(blue)(11)) = color(red)(-9/53)(x + color(blue)(23))
We can also solve the first equation for y to transform the equation to 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(blue)(2) = (color(red)(-9/53) xx x) - (color(red)(-9/53) xx color(blue)(30))
y - color(blue)(2) = -9/53x - (-270/53)
y - color(blue)(2) = -9/53x + 270/53
y - color(blue)(2) + 2 = -9/53x + 270/53 + 2
y - 0 = -9/53x + 270/53 + (53/53 xx 2)
y - 0 = -9/53x + 270/53 + 106/53
y = color(red)(-9/53)x + color(blue)(376/53)
