How do you write an equation of a line given (5, -2); (-16, 4)?

Question

Nathan Axel · Accepted Answer

See a solution process below:

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)(4) - color(blue)(-2))/(color(red)(-16) - color(blue)(5)) = (color(red)(4) + color(blue)(2))/(color(red)(-16) - color(blue)(5)) = 6/-21 = -2/7# We can now use the point-slope formula to write an equation for the line. 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)(-2/7)(x - color(blue)(5))# #(y + color(blue)(2)) = color(red)(-2/7)(x - color(blue)(5))# We can also substitute the slope we calculated and the values from the second point in the problem giving: #(y - color(blue)(4)) = color(red)(-2/7)(x - color(blue)(-16))# #(y - color(blue)(4)) = color(red)(-2/7)(x + color(blue)(16))#

Julia Adams · Answer

undefined To write the equation of a line given two points, you can follow these steps:

1. Find the slope $ m $ using the formula:
$$ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} $$

2. Use one of the given points and the slope to write the equation in point-slope form:
$$ y - y_1 = m(x - x_1) $$

3. If necessary, simplify the equation into slope-intercept form:
$$ y = mx + b $$

Using the points (5, -2) and (-16, 4):

1. Calculate the slope:
$$ m = \frac{{4 - (-2)}}{{-16 - 5}} = \frac{{4 + 2}}{{-16 - 5}} = \frac{6}{-21} = -\frac{2}{7} $$

2. Choose one of the points, let's say (5, -2), and substitute it into the point-slope form:
$$ y - (-2) = -\frac{2}{7}(x - 5) $$

3. Simplify if needed:
$$ y + 2 = -\frac{2}{7}(x - 5) $$
$$ y + 2 = -\frac{2}{7}x + \frac{10}{7} $$
$$ y = -\frac{2}{7}x + \frac{10}{7} - 2 $$
$$ y = -\frac{2}{7}x + \frac{10}{7} - \frac{14}{7} $$
$$ y = -\frac{2}{7}x - \frac{4}{7} $$

So, the equation of the line passing through the points (5, -2) and (-16, 4) is $ y = -\frac{2}{7}x - \frac{4}{7} $.