Given two points

Question

Write the equation of the line, in standard form, using the following information:

Point 1: (5,−7)

Point 2: (−1,2)

Answer 1

The slope of the line through the two points (x1,y1)=(5,-7) and (x2-y2)=( -1, 2) has a slope of (-7 -2)/(5 +1) = -9/6 = -3/2

Using the point slope equation: y - y1 = m(x - x1)

Then y - (-7) = (-3/2)(x - 5) y+7 = -1.5x + 7.5 Final: y = -1.5x +0.5

Answer 2

Standard form looks like the following:

Ax + By = C

Where A, B, and C are numbers. X and y stay variables (meaning they just stay as x and y).

This problem is actually pretty tricky, because you have to build the equation from scratch! Usually, the easiest equation to build from scratch is not standard form. I think the easiest equation to build from scratch is point-slope form, which we can then turn into standard form!

Making An Equation in Point-Slope Form

There are a couple ways to do this. Maybe you want to do it by graphing, if you're a visual learner. Or maybe you want to use some formulas.

The first formula you might want to use is the slope formula, to figure out the slope of the line. The slope formula needs 2 points, which is perfect, because that's what we have here.

[I won't go into how to use the slope formula, because I assume you already know how!]

After using the slope formula, I found that the slope of the line is -1.5.

Now that we have the slope of the line, we are ready to plug this into point-slope form. Point-slope form is the following:

y - y1 = m (x - x1)

Where m is the slope, x1 is the x-coordinate of a point, and y1 is the y-coordinate of the point. X and y are just our variables, so they stay x and y.

Let's pick one of the points we were given, it doesn't matter which one. I'll pick (-1, 2). Now, I'll plug everything into point slope form:

y - 2 = -1.5 (x + 1)

Turning Point-Slope Form into Standard Form

Now we just need to do some Algebra to make the equation look like Ax + By = C.

We start with:

y - 2 = -1.5 (x + 1)

Then we will multiple the -1.5 using the distributive property

y - 2 = -1.5 x - 1.5

Then we will add 2 to both sides to isolate the y.

y = -1.5x + .5

Then we will add 1.5x to both sides to put x and y on the same side of the equation.

1.5x + y = .5

Now this looks like standard form! We just need to make sure there are no decimals or fractions, because A, B, and C must all be integers. So let's multiple both sides by 2.

3x + 2y = 1

We did it!

Answer 3

Using point-slope form y2 - y1 = m (x2 - x1), we determine the slope of the line to be

2 + 7 = m (-1 - 5)

m = -3/2

After plugging point 1 into the point-slope form, we get

y + 7 = -3/2 (x - 5)

which simplifies to

y = -3x/2 + 1/2