Two Variables, One Line
Solutions and the line
Find solutions of a linear equation and see that they make a straight line.
A solution is an ordered pair (x, y) that makes the equation true. (2, 3) solves 3x + 2y = 12 because 6 + 6 = 12. Order matters: x comes first.
There are infinitely many solutions. Choose any value of x, and solve for y. Every choice gives another solution.
Plot the solutions: they all lie on one straight line, the graph of the equation. A point on the line is a solution; a point off the line is not.
Quick points to draw the line:
- put x = 0 to find where it crosses the y-axis;
- put y = 0 to find where it crosses the x-axis.
For 4x + 3y = 12 these are (0, 4) and (3, 0).
Switch c off: if c = 0, the line passes through the origin, because (0, 0) fits ax + by = 0. If c is not 0, it does not.
Finding a missing number. If (–3, 4) solves 3ax + 4y = –2, substitute: –9a + 16 = –2, so a = 2.
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