Co-ordinate Geometry
278. The Variable Boss · Who leads and who follows in a linear equation?
Point P always stays on the line y = 2x - 2.
The graph of a linear equation y = mx + c is a straight line, and every point on it satisfies the equation. For y = 2x − 2, a point P stays on the line: its y-coordinate is always twice its x-coordinate minus 2.
What this lesson covers
Try to break it
Drag P along the line. As x changes, y adjusts to keep P on the line — y is the dependent variable here. Slide P vertically and x adjusts instead — either variable can be the "boss". Try to drag P off the line; the constraint forces it back.
How you build it
Plot the line y = 2x - 2.
- Point tool: plot A at (0, −2) — where the line meets the y-axis (set x = 0 in y = 2x − 2).
- Point tool: plot B at (1, 0) — where the line meets the x-axis (set y = 0, so x = 1).
- Line tool: click A, then B — the straight line through the two intercepts is y = 2x − 2.
The proof, step by step
Prove that point P stays on the line y = 2x − 2.
- In y = 3x + 1, y is the dependent variable because it is isolated as the subject.
- In x = 4y - 5, x is the dependent variable because it is the subject of the formula.
- The independent variable is the one you can choose or change freely.
- The dependent variable's value depends on the value of the independent variable.
Worked example
In the equation 2x - 5y = 10, if y is expressed as the subject of the formula, which variable is the dependent variable?
Rearrange for y: 5y = 2x - 10 ⇒ y = (2/5)x - 2. Since y is isolated as the subject, y is the dependent variable.
- x
- y — correct
- 10
- 2