Pair of Linear Equations in Two Variables
A pair of linear equations
Tell from the coefficients whether two equations have one solution, none or many.
Start with one equation: x + y = 2. Which numbers fit? (0, 2), (2, 0), (1, 1), (4, –2), (100, –98) … there is no end. One equation in two variables has infinitely many solutions, and on a graph they all lie on one straight line.
A pair means two equations at once. A solution of the pair must fit both. On the graph, that is a point lying on both lines.
Real life gives pairs. Akhila at the fair: x Giant Wheel rides at ₹3, y Hoopla games at ₹4, and she played Hoopla half as many times as she rode. So y = ½x and 3x + 4y = 20.
Two lines on a page can only do three things:
| The lines | Solutions | The pair is called |
|---|---|---|
| cross at one point | exactly one | consistent |
| are parallel | none | inconsistent |
| lie on top of each other | infinitely many | dependent (and so consistent) |
You can tell which without drawing. First write both equations in the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, with everything on the left. (7x = 3y + 2 becomes 7x – 3y – 2 = 0, so c₁ = –2.) Then compare the ratios:
| Ratios | Lines |
|---|---|
| a₁/a₂ ≠ b₁/b₂ | cross: one solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | the same line: infinitely many |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | parallel: none |
When there is a letter in it. For kx + 3y = –2 and 2x + 4y + 5 = 0, you can't say how many solutions until you know k. For one solution you need k/2 ≠ 3/4, so k ≠ 3/2: the answer is 'every value of k except 3/2'.
Always put your k back in and check all three ratios. Sometimes the value you find makes the lines the same instead of parallel, and then the honest answer is 'no such value'.
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