How do you write an equation of a line going through (-5,-2), (-3,8)?

1 Answer
Mar 7, 2017

#(y + color(red)(2)) = color(blue)(5)(x + color(red)(5))#

Or

#(y - color(red)(8)) = color(blue)(5)(x + color(red)(3))#

Or

#y = color(red)(5)x + color(blue)(23)#

Explanation:

First, we need to 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)(8) - color(blue)(-2))/(color(red)(-3) - color(blue)(-5)) = (color(red)(8) + color(blue)(2))/(color(red)(-3) + color(blue)(5)) = 10/2 = 5#

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))#

Where #color(blue)(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)(-2)) = color(blue)(5)(x - color(red)(-5))#

#(y + color(red)(2)) = color(blue)(5)(x + color(red)(5))#

Or, we can substitute the slope we calculated and the second point from the problem giving:

#(y - color(red)(8)) = color(blue)(5)(x - color(red)(-3))#

#(y - color(red)(8)) = color(blue)(5)(x + color(red)(3))#

Or, we can solve for #y# and put the equation 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)(8) = (color(blue)(5) xx x) + (color(blue)(5) xx color(red)(3))#

#y - color(red)(8) = 5x + 15#

#y - color(red)(8) + 8 = 5x + 15 + 8#

#y - 0 = 5x + 23#

#y = color(red)(5)x + color(blue)(23)#