Calculus problem

Question

Find the standard form equation of the line passing through (-10,-2) and (4,-8). Make sure that your final answer has only integer values.

Answer 1

slope is rise over run, or the ratio of the change in y values over the change in x values = m

or m=(-8+2)/(4+10)=--6/14=-3/7

replace either of the points, such as (4,-8) by (x,y), where 4 appeared replace it by x, where -8 appeared replace it by y

then m=(y+2)/(x+10)=-3/7

cross multiply to get: 7(y+2)=-3(x+10)

expand to get 7y+14 = -3x-30

rewrite and collect the constant terms

3x+7y=-30-14=-44

3x+7y=-44

check the answer, plugging in each of the two points (-10,-2), (4,-8)

try the first point first

3(-10)-7(-2)=-30+14=44

then try (4,-8)

2(4)+7(-8)=12-56=-44

It works. You can rewrite the equation 3x+7y=-44

as 3x+7y+44=0 if you want. Those are about the only ways to keep the equation with all integers

Rewrite it in point slope form, and you get fractions: y=-3x/7-44/7

You might try just sketching a rough graph of the two points and a line connecting them.

(-10,-2) is in Quadrant III not far from the x-axis, just 2 units down from it

(4,-8) is in Quadrant IV, to the right and down lower than the first point.

A line connecting them will cross the y-axis somewhere between -2 and -8, closer to the -8 though.

-44/7 is a little less than -6, so it looks right. The y intercept is -44/7

Answer 2

First of all, remember what the equation of a line is:

y = mx+b

Where:

m is the slope, and b is the y-intercept

Now, let's find the slope of the line

For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

m = y₂ – y₁/ x₂ – x₁

For the first point, (-10, 2), x₁=-10 and y₁=2. For the second point, (4,-8) x₂=4 and y₂=-8.

Now, just plug the numbers into the formula for m above, like this:

m = -8 – -2/4  – -10 = -8 +2/4+10 = -6/14 = -3/7

To find b, think about what your (x,y) points mean:

·       (-10,2). When x of the line is -10, y of the line must be 2.

·       (4,-8). When x of the line is 4, y of the line must be -8.

Now, we will solve for b by substituting in a point and the slope we just calculated. I am going to use the first point:

y = mx + b

-2 = -3/7(-10) + b

-2 = 30/7 + b

-2(7/30) = b

-7/15 = b

The answer so far is y = -3/7x + -7/15.

However, the problem asks for only integer values and I am not sure how to do this as I have been out of Calculus for a while and have forgotten this trick. I would suggest asking your teacher.