Step by step
The question
Question: graph −x + 3y ≤ 6.
The answer is not one point.
It is every point that makes this true.
That is a whole part of the grid.
1 Draw the line
First we draw the line where both sides are equal: −x + 3y = 6.
The sign has "or equal to".
So points on the line are answers too.
We draw it solid, like the filled circle on a number line.
On the x-axis, y = 0.
Put 0 in place of y.
3 times 0 is 0.
Adding 0 changes nothing.
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it on both sides, so they stay equal.
−1 divided by −1 is 1.
So just x is left.
Plus over minus is minus.
6 divided by −1 is −6.
So x = −6.
On the y-axis, x = 0.
Put 0 in place of x.
1 times 0 is 0.
0 plus 3y is just 3y.
3y means 3 times y.
We want just y.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just y is left.
6 divided by 3 is 2.
So y = 2.
So the line crosses the axes at (−6, 0) and (0, 2).
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put the two crossing points on the grid,
and join them: solid.
2 Test a point
The line cuts the grid in two sides.
One side is the answer.
We test one point: (0, 0).
Zeros are easy to work with.
Put x = 0 and y = 0 in.
Is 0 less than or equal to 6?
Yes, that is true.
So (0, 0) is an answer.
3 Shade
So we shade the side with (0, 0).
Every point there is an answer.
4 Check
Let's check with a point in the shaded part: (−2, −1).
Put x = −2 and y = −1 in.
True!
So the shading is right.
Let's look back.
First: draw the line.
Then: test a point.
Then: shade.
Last: check.