You have graphed lines like $y = 2x - 1$ in slope-intercept form. It is worth saying exactly what such a graph is, because the idea carries through the rest of algebra: the graph of an equation is a picture of all its solutions.
Why it works
A solution of $y = 2x - 1$ is a pair of numbers $(x, y)$ that makes it true, like $(3, 5)$, since $2(3) - 1 = 5$. The graph is drawn by plotting every such pair. So two things are true at once: every point on the line makes the equation true, and every pair that makes the equation true is a point on the line. A point off the line, like $(3, 4)$, fails the equation. This is also what function notation says: $f(3) = 5$ means $(3, 5)$ is on the graph of $f$.
Key idea
A point $(a, b)$ is on the graph of an equation exactly when substituting $x = a$ and $y = b$ makes the equation true. To check a point, substitute; do not trust your eyes.
Worked examples
Worked example: Check a point
Is $(4, -1)$ on the graph of $3x + 5y = 7$?
- Substitute $x = 4$ and $y = -1$: $3(4) + 5(-1) = 12 - 5 = 7$.
- $7 = 7$ is true.
Answer: Yes, $(4, -1)$ is on the graph.
Worked example: Find a missing coordinate
The point $(-2, k)$ is on the graph of $2x - 3y = 8$. Find $k$.
- The point makes the equation true: $2(-2) - 3k = 8$.
- $-4 - 3k = 8$, so $-3k = 12$ and $k = -4$.
Answer: $k = -4$
Common mistake
Checking a point by looking at a picture. On a graph, $(3, 4.9)$ and $(3, 5)$ look the same, but only one is on the line $y = 2x - 1$. Substitute the coordinates; the equation gives a yes or no answer that the eye cannot.