How do you write an equation for the line through (6,8) and (2,-10)?

Question

How do you write an equation for  the line through (6,8) and (2,-10)?

Hazel Anglin · Accepted Answer

#y - 8 = 9/2(x - 6)# or #y = 9/2x - 19#

To write an equation for a line through two points we need to use a two step process.

Step 1) Find the slope.

Step 2) Using the slope, one of the points and the point-slope formula determine the equation.

The slope can be found by using the formula: #color(red)(m = (y_2 - y_1)/(x_2 - x_1)# Where #m# is the slope and #(color(red)((x_1, y_1)))# and #(color(red)((x_2, y_2)))# are the two points.

Substituting the points we are given in this problem the slope is:

#m = (-10 - 8)/(2 - 6)#

#m = (-18)/(-4)#

#m = (-2 xx 9)/(-2 xx 2)#

#m = (-2)/(-2) xx 9/2#

#m = 1 xx 9/2#

#m = 9/2#

Now we can use one of the points we were given, the slope we determined and the point-slope formula to find the equation for the line.

The point-slope formula states: #color(red)((y - y_1) = m(x - x_1))# Where #color(red)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

#y - 8 = 9/2(x - 6)#

If we want to put this into slope-intercept form we can solve for #y#:

#y - 8 = 9/2x - (9/2 xx 6)#

#y - 8 = 9/2x - (9 xx 3)#

#y - 8 = 9/2x - 27#

#y - 8 + 8 = 9/2x - 27 + 8#

#y - 0 = 9/2x - 19#

#y = 9/2x - 19#

Hazel Anglin · Answer

undefined To write an equation for the line through (6,8) and (2,-10), you can use the point-slope form of a linear equation:

y - y₁ = m(x - x₁)

Where (x₁, y₁) is one point on the line and m is the slope of the line.

First, calculate the slope using the given points:

m = (y₂ - y₁) / (x₂ - x₁)
m = (-10 - 8) / (2 - 6)
m = (-18) / (-4)
m = 4.5

Next, choose one of the points to substitute into the equation. Let's use (6,8):

x₁ = 6
y₁ = 8

Plug the slope and the coordinates of the point into the point-slope form:

y - 8 = 4.5(x - 6)

Now, simplify the equation:

y - 8 = 4.5x - 27

Add 8 to both sides:

y = 4.5x - 19

So, the equation for the line through (6,8) and (2,-10) is y = 4.5x - 19.