What is the equation of the line that has a slope of #-3/4# and passes through point (8,2) ?

Equation of the line with slope of #-3/4# and passes through #(8,2)#.

Use the point slope formula where #m=# slope
and #(x_1, y_1)# is a point.

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

#y-2= -3/4 (x-8)#

This is the equation in point slope form.

If you'd like to express the equation in slope intercept form, distribute the #-3/4# across the terms in the parentheses.

#y-2= -3/4x -3/4 * -8#

#y-2 = -3/4 x +6#

#color(white)(a)+2color(white)(aaaaaaaa)+2color(white)(aaa)#Add 2 to both sides

#y = -3/4 x +8#

First, let's try to connect the question to some concepts that we may be familiar with already. It's important to note that #y = mx + b#, where #m# is defined as the #\text{slope}# and #b# is defined as the #\text{y-intercept}#. This also happens to be the #\text{slope-intercept form}# for the equation of a line! We can use this equation to find our answer.

This question gives you the following information:

#\text{1}#. #m = -\frac{3}{4}#

#\text{2}#. Coordinates #(8, 2)#

So, how can we use this information to find our answer?

Well, if we use (1), we can rewrite our equation #y = mx + b# as the following:

#y = -\frac{3}{4}x + b#.

Note that all I did was replace the #m# variable with the actual number. Well, we still need to find our #b# term! We can use (2) to accomplish just that. Input the coordinates #(8, 2)# into our equation. Since #x = 8# and #y = 2#, we have

#2=-\frac{3}{4} \cdot 8 + b = -\frac{24}{4} + b#

Luckily for us, 4 divides into -24 perfectly times, leaving us with -6

#2 = -6 + b#

Adding #+6# to both sides (you try this step and see what happens!), we obtain

#8 = b#.

Hence, our #\text{equation of the line}# in #\text{slope-intercept form}# is #y = -\frac{3}{4}x + 8#. We can also multiply each term by 4 to obtain

#4y = -3x + 32# which becomes #3x + 4y = 32# after we add #3x# to both sides of the equation.

#\text{CHECK}#

It's important to note that the equation of the line characterizes the line; that is, every x-value has a unique y-value. We know that when #x = 8#, we should get #y = 2# since #(8, 2)# is a coordinate on the line. Well, let's plug in #x = 8# into our solution and, if we get y = 2, we definitely know we're correct.

Using the slope-intercept form and inputting #x = 8#, we obtain

#y = -\frac{3}{4} \cdot8 + 8 = -6 + 8 = 2#.

So, when #x = 8#, we get #y = 2#. This verifies that our solution is correct.

• htriber