Count the shared dots
A pair can have one, none or endlessly many common solutions.
Two equations, one answer?
Take two equations with two letters: x − 2y = 0 and 3x + 4y − 20 = 0. We want numbers (x, y) that make both true.
How many such pairs of numbers can there be?
What this lesson covers
The idea
A pair of linear equations that has a solution is consistent and one with no solution is inconsistent; a pair of equivalent equations has infinitely many common solutions and is called dependent, which is always consistent.
Two equations, one answer?
Take two equations with two letters: x − 2y = 0 and 3x + 4y − 20 = 0. We want numbers (x, y) that make both true.
How many such pairs of numbers can there be?
- Always exactly one
- One, none, or endlessly many
- Always endlessly many
Light up the dots
Each equation lights up the whole-number points (dots) that make it true: blue for (1), orange for (2). A dot lit by both turns green. Open all three pairs, and move the cursor to test any point.
Consistent, inconsistent, dependent
A pair of linear equations that has a solution is consistent. A pair that has no solution is inconsistent. A pair of equivalent equations has infinitely many common solutions. It is called dependent, and it is always consistent.
In the dots: one green dot is a consistent pair. No green dot is an inconsistent pair. Every dot green is a dependent pair (and consistent too).
Why does pair 3 have no common solution? Divide its second equation by 2: it becomes x + 2y − 6 = 0, so x + 2y = 6. But the first equation says x + 2y = 4. One number cannot be both 4 and 6, so no dot is ever shared.
And pair 2? 4x + 6y − 18 = 0 is just 2 × (2x + 3y − 9) = 0. It is the same equation in disguise, so every solution of one is a solution of the other.
Notes
A pair with a solution is consistent; with no solution, inconsistent. A pair of equivalent equations has infinitely many common solutions: it is dependent, and always consistent.
Check yourself
A pair of equations has no common solution. It is called…
2x + 3y − 9 = 0 and 4x + 6y − 18 = 0: the second is 2 times the first. This pair is…
How many common solutions do x + 2y − 4 = 0 and 2x + 4y − 12 = 0 have?
Answer: 0
The second equation says x + 2y = 6 and the first says x + 2y = 4. They cannot both hold, so there is no common solution: the pair is inconsistent.
True or false? A dependent pair is always consistent.
- inconsistent — correct. Yes! No solution means inconsistent.
- consistent. A consistent pair has at least one solution.
- dependent. A dependent pair has infinitely many solutions.
- dependent, and so also consistent — correct. Yes! The same equation in disguise: endlessly many common solutions.
- inconsistent. They share every solution, so there are plenty.
- consistent with exactly one solution. The second equation is just the first doubled, so they share every solution, not only one.
- True: it has solutions, infinitely many — correct. Yes! Consistent only asks for at least one solution.
- False: dependent means no solution. Dependent means infinitely many solutions, and that is more than none.