Drawing a relationship

A rule connecting x and y is not a number — it is a whole line of points at once.

Every lesson so far has plotted one point at a time. This one plots a rule.

A fixed line of points, all satisfying $y = x + 1$, and one draggable point P. A live reading checks whether P’s own coordinates satisfy the same rule.

Try to find a spot for P that’s close to the line but doesn’t count — the reading will show exactly how far off the rule it is, even by a fraction.

A rule connecting two varying quantities — here, $x$ and $y$ — is a relationship. $x$ is not one fixed unknown to solve for here, the way it is in an equation like $2x + 1 = 7$; it is a number that varies, with $y$ varying alongside it according to the rule.

One rule, tested against every point at once

$y = x + 1$ is not a statement about one pair of numbers. It is a test any pair can be run through: put in a value for $x$, and the rule says exactly what $y$ has to be for that pair to belong to the relationship. Every point on the fixed line passed that test; P only joins them when it does too.

This is the bridge the rest of algebra crosses on: an equation with two letters in it is not something to “solve” for a single answer, it describes a whole collection of pairs. The next unit gives that collection a name (a function), a picture (a graph), and a shape (a straight line, from $y = mx + c$), but the idea itself is already complete here.

Testing points against the rule

Does the point $(4, 5)$ satisfy $y = x + 1$?

$x + 1 = 4 + 1 = 5$, and that is exactly the point’s $y$. Yes, it satisfies the relationship.

Does the point $(3, 3)$ satisfy $y = x + 1$?

$x + 1 = 3 + 1 = 4$, but the point’s $y$ is 3, not 4. No — it is 1 short.

Find a point, other than $(2, 3)$, that satisfies $y = x + 2$.

Any pair where $y$ is exactly $x + 2$ works, for example $(0, 2)$: $x + 2 = 0 + 2 = 2$, which matches.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.