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

Dec 17, 2017

The standard form of the equation for the given line:

$3 x - 2 y = - 10$

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

$y = m x + b$

1) First find slope
slope $= m = \frac{y - y '}{x - x '}$

Let $\left(0 , 5\right)$ be $\left(x ' , y '\right)$
But you can make either point be $\left(x ' , y '\right)$ 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")

$m = \frac{2 - 5}{- 2 - 0}$

$m = \frac{- 3}{- 2}$

$m = \frac{3}{2}$

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

$y = \left(\frac{3}{2}\right) x + b$

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2) Now find b

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

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

$m = \frac{3}{2}$ (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 $\left(0 , 5\right)$ 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 $b$

$y = m x + b$

$5 = \left(\frac{3}{4}\right) \cdot 0 + b$

$5 = 0 + b$

$5 = b$ $\leftarrow$ $b$ is the $y$ intercept

3) Now you can write the whole equation

$y = \left(\frac{3}{2}\right) x + 5$

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The question wants this equation expressed in standard form.

Standard form is
$a x + b y = c$
where a is a positive whole integer

$y = \left(\frac{3}{2}\right) x + 5$

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
$2 y = 3 x + 10$

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

3) Subtract $2 y$ from both sides
$3 x - 2 y = - 10$ $\leftarrow$ standard form