How do you find a standard form equation for the line with (0, -10) and (-4, 0)?

Question

Avery Alexander · 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)(0) - color(blue)(-10))/(color(red)(-4) - color(blue)(0)) = (color(red)(0) + color(blue)(10))/(color(red)(-4) - color(blue)(0)) = 10/-4 = -5/2# Because the first point in the problem has a value of #0# for #x# we know this is the #y#-intercept. The y-intercept is therefore: #color(blue)(-10)# We can now write and 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. Substituting the slope we calculated and the #y#-intercept we determined gives: #y = color(red)(-5/2)x + color(blue)(-10)# #y = color(red)(-5/2)x - color(blue)(10)# We need to convert this equation to the Standard Form for Linear equations. The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)# Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1 First, we can add #5/2x# to each side of the equation to put the #x# and #y# variables on the left side of the equation as required by the Standard formula: #5/2x + y = 5/2x + color(red)(-5/2)x - color(blue)(10)# #5/2x + y = 0 - color(blue)(10)# #5/2x + y = -10# We can now multiply each side of the equation by #color(red)(2)# to eliminate the fraction as required by the Standard formula: #color(red)(2)(5/2x + y) = color(red)(2) xx -10# #(color(red)(2) xx 5/2x) + (color(red)(2) xx y) = -20# #(cancel(color(red)(2)) xx 5/color(red)(cancel(color(black)(2)))x) + 2y = -20# #color(red)(5)x + color(blue)(2)y = color(green)(-20)#

Kennedy Adams · Answer

undefined To find the standard form equation for the line passing through the points (0, -10) and (-4, 0), you can use the point-slope form of the equation of a line:

$$ y - y_1 = m(x - x_1) $$

where $ (x_1, y_1) $ is one of the given points and $ m $ is the slope of the line. First, calculate the slope using the formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Substitute the coordinates of the given points:

$$ m = \frac{0 - (-10)}{-4 - 0} = \frac{10}{-4} = -\frac{5}{2} $$

Next, choose one of the given points (either (0, -10) or (-4, 0)) and substitute its coordinates and the slope into the point-slope form equation. Let's use the point (0, -10):

$$ y - (-10) = -\frac{5}{2}(x - 0) $$
$$ y + 10 = -\frac{5}{2}x $$

Now, rearrange the equation to the standard form (Ax + By = C):

$$ \frac{5}{2}x + y + 10 = 0 $$

Multiply both sides of the equation by 2 to clear the fraction:

$$ 5x + 2y + 20 = 0 $$

This is the standard form equation for the line passing through the points (0, -10) and (-4, 0).