Question

How do you find a standard form equation for the line with (-2,2) and (0,5)?

Answer 1

The standard form of the equation for the given line:

`3x - 2y = - 10`

Explanation:

One way to do this is to find the slope-intercept form of the equation, and then turn that into standard form.

Another way is to use the point-slope form and turn that into standard form.

Using the slope-intercept form of the equation

`y = mx + b`

1) First find slope
slope `= m = (y- y')/(x - x')`

Let `(0,5)` be `(x',y')`
But you can make either point be `(x',y')` and it will come out the same.
I just picked that one because I thought it looked easier to subtract.

Sub in the given values for the variables in the formula
`m` = `(y - y') / (x - x")`

`m =(2 - 5) / (-2-0)`

`m = (-3) / (-2)`

`m = 3/2`

So far, you can write the partial slope intercept formula as

`y = (3/2)x + b`

~ ~ ~ ~ ~ ~ ~ ~ ~

2) Now find b

The formula has 4 unknowns, but you already know 3 of them

`y = 5` (from the given ordered pair)

`m = 3/2` (which you just found)

`x = 0` (from the same ordered pair)

`b` ---- Find `b`

NOTE: You can tell that `5` is the `y` intercept because the point `(0,5)` is the point where `x` is zero.
This is the place where the line crosses the y axis.

But if you didn't see that, here is how to calculate `b`

Sub in the values into the formula and solve for `b`

`y = mx + b`

`5 = (3/4)*0 + b`

`5 = 0 + b`

`5 = b` `larr` `b` is the `y` intercept

3) Now you can write the whole equation

`y = (3/2)x + 5`

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The question wants this equation expressed in standard form.

Standard form is
`ax + by = c`
where a is a positive whole integer

Start with the slope intercept form.

`y = (3/2)x + 5`

4) Now put the formula in standard form.

1) Clear the fraction by multiplying every term on both sides by `2` and letting the denominator cancel
`2y = 3x + 10`

2) Subtract `10` from both sides
`2y - 10 = 3x`

3) Subtract `2y` from both sides
`3x - 2y = - 10` `larr` standard form