Question

How do you graph `x+y=5` using intercepts?

Answer 1

1. Get the `x`-intercept by setting `y=0` and solving for `x`.
2. Get the `y`-intercept by setting `x=0` and solving for `y`.

Graph these two points, and connect them.

Explanation:

An intercept is simply a point where a line (or any function) crosses an axis. At such a point, the non-axis coordinate is 0. (e.g. all points on the `x`-axis have no displacement in the `y` direction, so the `y`-coordinate is `0`.)

We can use this to help us get two points that are on the line, graph those two points, and finally connect them by drawing a line through them.

The `x`-intercept occurs when `y=0`. To find this intercept, we just need to plug in 0 for `y` in our line equation:

`color(white)(=>)x+y=5 " @ "y=0`
`=>x+0=5`
`=>xcolor(white)+color(white)0=5`

So, when `y=0`, we have `x=5`, meaning `(5,0)` is a point on our line, and it is the `x`-intercept.

Similarly, to find the `y`-intercept, we let `x=0` and solve for `y`:

`color(white)(=>)x+y=5 " @ "x=0`
`=>0+y=5`
`=>color(white)x color(white)+ y=5`

So when `x=0`, `y=5`, and thus `(0,5)` is our `y`-intercept.

Finally, we graph these two points, and connect them with a line:

graph{(x+y-5)((x-5)^2+y^2-0.06)(x^2+(y-5)^2-0.06)=0 [-8.325, 11.68, -2.3, 7.7]}

Bonus:

For a linear equation of the form `x+y=k` for some number `k`, the equation is saying "two numbers sum to `k`". From this, it is easy to see that when either number is 0, the other one must be `k`, so that the sum will be `k`. This means the intercepts will always be `(0,k)` and `(k,0)`.