Question

How do you graph `54x - 42y = 37,800`?

Answer 1

• Reduce the coefficients to lowest terms

• Write the equation in slope-intercept form

• Set up a graph labeled in hundreds, not ones

• Write the slope as `(900)/(700)`

Explanation:

Graph
`54x−42y=37,800`

There are two things that are making this problem difficult:
• The numbers are too large.
• The equation is not in slope-intercept form

Reduce the coefficients to their lowest terms

`1)` Divide each coefficient by 6
`9x - 7y = 6300`

Write the equation in slope-intercept form

1) Subtract `9x` from both sides to isolate the `-7y` term
`- 7y = -9x +6300`

`2)` Divide all the terms on both sides by `-7` to isolate `y`
`y = (9)/(7)x - 900`

Set up a graph that is convenient to use

There is no reason why the squares on a sheet of graph paper have to be labeled as ones
`0, 1, 2, 3,` etc.

For this problem, label the squares in hundreds instead of ones
`0, -100, -200, -300`, etc.

Now it's easy to find the place to plot the `y` intercept, down at `-900`.

But now there's another problem.

The slope of `(9)/(7)` is written in numbers that are too small to be graphed on such a big scale.

So write `(9)/(7)` as `(900)/(700)`

Now the equation is

`y = ((900)/(700))x - 900`

This is easy to graph by hand (or even roughly sketch) on a grid that is labeled in hundreds
graph{y = ((900)/(700))x - 900 [-1243, 1317, -1171, 109]}

Answer 2

Find the two intercepts and draw a line through them.

Explanation:

The equation looks more uncomfortable than it is,
Simplify by dividing each term by the `HCF` which is `6`

`54x-42y = 37800`

`9x -7y =6300`

You can use the `x- and y-` intercepts to draw the graph quickly and easily.

All along the `y-` axis, `x=0` and all along the `x-` axis, `y=0`

To find the `y-` intercept, set `x=0` and solve for `y`

`9(0) -7y =6300`

`-7y =6300`

`y = -900" "larr` the `y-` intercept

To find the `x-` intercept, set `y=0` and solve for `x`

`9x -7(0) =6300`

`9x =6300`

`x = 700" "larr` the `x-` intercept

Plot the two intercepts and draw a line through them.

graph{y =9/7x-900 [-784, 1776, -1142, 138]}