What "on the line" means
The graph of an equation is just the collection of every point whose coordinates satisfy it. Nothing more mysterious than that. If a point sits on the line, its \(x\) and \(y\) obey the rule. If it does not, they break the rule.
- Lies on the line
- A point \((a, b)\) lies on a line when substituting \(x = a\) and \(y = b\) makes the equation a true statement, with the two sides equal.
The substitution test
To test a point, replace \(x\) and \(y\) in the equation with the point's coordinates and check whether both sides come out equal. If they do, the point is on the line. If they do not, it is off the line.
Substitute both coordinates
Put in the \(x\)-value and the \(y\)-value, then compare. For a form like \(2x + 3y = 12\), work out the left side and check it really equals the right side.
Worked example
Does the point \((2, 7)\) lie on the line \(y = 3x + 1\)?
Finding a missing coordinate
The same idea fills in a gap. If you know a point lies on a line but one coordinate is unknown, substitute what you have and solve for the rest.
The point \((4, k)\) lies on \(y = 2x - 3\). Find \(k\).
Check yourself
1. Does \((3, 10)\) lie on \(y = 4x - 2\)? +
Substitute \(x = 3\): \(4(3) - 2 = 10\). This equals the \(y\)-value \(10\), so yes, it lies on the line.
2. Does \((0, 0)\) lie on \(y = 2x + 1\)? +
Substitute \(x = 0\): \(2(0) + 1 = 1\). The point's \(y\)-value is \(0\), and \(1 \neq 0\), so no, it does not lie on the line.
3. The point \((5, k)\) lies on \(y = -x + 9\). Find \(k\). +
Substitute \(x = 5\): \(k = -(5) + 9 = 4\). So \(\mathbf{k = 4}\).
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.