How do you find the equation given A(-2, 1) and B(3, 7)?

Question

Genesis Allen · Accepted Answer

See a solution process below:

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)(7) - color(blue)(1))/(color(red)(3) - color(blue)(-2)) = (color(red)(7) - color(blue)(1))/(color(red)(3) + color(blue)(2)) = 6/5# 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))# 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 values from the first point in the problem gives: #(y - color(red)(1)) = color(blue)(6/5)(x - color(red)(-2))# #(y - color(red)(1)) = color(blue)(6/5)(x + color(red)(2))# We can also substitute the slope we calculated and the values from the second point in the problem giving: #(y - color(red)(7)) = color(blue)(6/5)(x - color(red)(3))# We can now solve this equation 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)(7) = (color(blue)(6/5) xx x) - (color(blue)(6/5) xx color(red)(3))# #y - color(red)(7) = 6/5x - 18/5# #y - color(red)(7) + 7 = 6/5x - 18/5 + 7# #y - 0= 6/5x - 18/5 + (7 xx 5/5)# #y = 6/5x - 18/5 + 35/5# #y = color(red)(6/5)x + color(blue)(17/5)#

Savannah Anderson · Answer

undefined To find the equation of a line given two points, you can use the point-slope form of the equation, which is $ y - y_1 = m(x - x_1) $, where $ m $ is the slope of the line and $ (x_1, y_1) $ is one of the given points.

First, calculate the slope using the formula $ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} $, where $ (x_1, y_1) $ and $ (x_2, y_2) $ are the coordinates of the two points A and B.

Then, choose one of the points, let's say A(-2, 1), plug its coordinates and the calculated slope into the point-slope form equation to find the equation of the line.

Using point A(-2, 1) and B(3, 7):

Slope $ m = \frac{{7 - 1}}{{3 - (-2)}} = \frac{6}{5} $

Now, plug the slope and the coordinates of point A into the point-slope form:

$ y - 1 = \frac{6}{5}(x - (-2)) $

Simplify:

$ y - 1 = \frac{6}{5}(x + 2) $

$ y - 1 = \frac{6}{5}x + \frac{12}{5} $

$ y = \frac{6}{5}x + \frac{12}{5} + 1 $

$ y = \frac{6}{5}x + \frac{12}{5} + \frac{5}{5} $

$ y = \frac{6}{5}x + \frac{17}{5} $

So, the equation of the line passing through points A(-2, 1) and B(3, 7) is $ y = \frac{6}{5}x + \frac{17}{5} $.