What is the point-slope form of the three lines that pass through (1,-2), (5,-6), and (0,0)?

Question

Ethan Adkins · Accepted Answer

See a solution process below:

First, let's name the three points. #A# is #(1, -2)#; #B# is #(5, -6)#; #C# is #(0,0)# First, let's find the slope of each 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. Slope A-B: #m_(A-B) = (color(red)(-6) - color(blue)(-2))/(color(red)(5) - color(blue)(1)) = (color(red)(-6) + color(blue)(2))/(color(red)(5) - color(blue)(1)) = -4/4 = -1# Slope A-C: #m_(A-C) = (color(red)(0) - color(blue)(-2))/(color(red)(0) - color(blue)(1)) = (color(red)(0) + color(blue)(2))/(color(red)(0) - color(blue)(1)) = 2/-1 = -2# Slope B-C: #m_(A-B) = (color(red)(0) - color(blue)(-6))/(color(red)(0) - color(blue)(5)) = (color(red)(0) + color(blue)(6))/(color(red)(0) - color(blue)(5)) = 6/-5 = -6/5# 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. We can substitute each of the slopes we calculated and one point from each line to write an equation in point-slope form: Line A-B: #(y - color(blue)(-2)) = color(red)(-1)(x - color(blue)(1))# #(y + color(blue)(2)) = color(red)(-1)(x - color(blue)(1))# Or #(y + color(blue)(2)) = color(red)(-)(x - color(blue)(1))# Line A-C: #(y - color(blue)(-2)) = color(red)(-2)(x - color(blue)(1))# #(y + color(blue)(2)) = color(red)(-2)(x - color(blue)(1))# Line B-C: #(y - color(blue)(-6)) = color(red)(-6/5)(x - color(blue)(5))# #(y + color(blue)(6)) = color(red)(-6/5)(x - color(blue)(5))#

Jace Anderson · Answer

undefined To find the point-slope form of a line passing through two points (x₁, y₁) and (x₂, y₂), you use the formula:

$$y - y₁ = m(x - x₁)$$

where $m$ is the slope of the line.

1. Find the slope $m$ using two points: $(1,-2)$ and $(5,-6)$.
$$m = \frac{y₂ - y₁}{x₂ - x₁} = \frac{-6 - (-2)}{5 - 1} = \frac{-4}{4} = -1$$

2. Now, plug in one of the points and the slope into the point-slope form.
$$y - (-2) = -1(x - 1)$$
$$y + 2 = -x + 1$$

3. Simplify the equation:
$$y = -x - 1$$

So, the point-slope form of the line passing through (1,-2) and (5,-6) is $y = -x - 1$.

You repeat the same process for the other two points to get the equations of the other two lines.