How do you write an equation of the line given (5, 2) and (-7, 3)?

Question

Isaiah Anthony · Accepted Answer

#(y - color(red)(2)) = color(blue)(-1/12)(x - color(red)(5))#

Or

#(y - color(red)(3)) = color(blue)(-1/12)(x + color(red)(7))#

Or

#y = -1/12x + 29/12# Given two points we can use the point-slope formula to find an equation for a line. First, determine the slope. 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)(3) - color(blue)(2))/(color(red)(-7) - color(blue)(5))# #m = 1/-12 = -1/12# 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. We can use the first point and the slope we calculate to give: #(y - color(red)(2)) = color(blue)(-1/12)(x - color(red)(5))# We can also use the second point and the slope we calculate to give: #(y - color(red)(3)) = color(blue)(-1/12)(x - color(red)(-7))# #(y - color(red)(3)) = color(blue)(-1/12)(x + color(red)(7))# We can also solve this for #y# to give an equation in slope-intercept form: #y - color(red)(3) = (color(blue)(-1/12) xx x) + (color(blue)(-1/12) xx color(red)(7))# #y - color(red)(3) = color(blue)(-1/12)x - 7/12# #y - color(red)(3) + 3 = color(blue)(-1/12)x - 7/12 + 3# #y - 0 = color(blue)(-1/12)x - 7/12 + 36/12# #y = -1/12x + 29/12#

Eva Adamson · Answer

undefined To write the equation of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$, you can use the point-slope form:

$y - y_1 = \frac{{y_2 - y_1}}{{x_2 - x_1}}(x - x_1)$

Substitute the given points $(5, 2)$ and $(-7, 3)$ into the formula:

$y - 2 = \frac{{3 - 2}}{{-7 - 5}}(x - 5)$

Simplify:

$y - 2 = \frac{{1}}{{-12}}(x - 5)$

$y - 2 = -\frac{{1}}{{12}}(x - 5)$

To convert it to slope-intercept form $y = mx + b$, distribute and simplify:

$y - 2 = -\frac{{1}}{{12}}x + \frac{{5}}{{12}}$

$y = -\frac{{1}}{{12}}x + \frac{{29}}{{12}}$