Systems of Equations • Topic 3 of 4

No / Infinite Solutions

Not every pair of equations has one neat answer. Geometrically, two linear equations are two lines: if the lines cross, there is exactly one solution; if they are parallel — same slope but a different intercept — they never meet, so there is no solution; if they are actually the same line written two ways, every point works and there are infinitely many solutions. Algebraically you can spot these without graphing. If eliminating a variable leaves a false statement like 0 = 6, the system has no solution. If it leaves a true statement like 0 = 0, one equation is a multiple of the other and there are infinitely many solutions. On the GRE this is exactly the reasoning behind "cannot be determined" in Quantitative Comparison: when a system has infinitely many solutions, a quantity built from x and y often is not fixed.

✅ Solved examples

1. How many solutions does x + y = 4 and 2x + 2y = 8 have?
The second is exactly 2× the first — same line. Infinitely many solutions.
2. How many solutions does x + y = 4 and 2x + 2y = 10 have?
Same left side scaled (2x + 2y = 8 from the first), but the constants clash (8 ≠ 10) — parallel lines. No solution.
3. Solve 3x − y = 5 and 6x − 2y = 7. How many solutions?
Double the first: 6x − 2y = 10, but the second says 7. 10 ≠ 7, contradiction. No solution.
4. For what value of k does kx + 2y = 6 and 4x + 2y = 6 have infinitely many solutions?
They must be the same line. The right sides and the y-terms already match, so we need kx = 4x ⇒ k = 4.

✏️ Practice — try these, take hints as needed

1. How many solutions does 2x + y = 3 and 4x + 2y = 6 have?
Is the second a multiple of the first?
Multiply the first by 2.
4x + 2y = 6 matches exactly.
Infinitely many
2. How many solutions does x − y = 1 and x − y = 4 have?
Same left side.
Constants 1 vs 4.
Parallel lines.
No solution
3. How many solutions does y = 2x + 1 and y = 3x − 2 have?
Compare slopes.
2 ≠ 3, lines cross.
They meet once.
Exactly one
4. For what k does 3x + ky = 9 and 3x + 6y = 9 have infinitely many solutions?
Same line needed.
x-terms and constants already match.
Set ky = 6y.
k = 6
5. How many solutions does 2x − 4y = 8 and x − 2y = 5 have?
Halve the first: x − 2y = 4.
Second says x − 2y = 5.
4 ≠ 5.
No solution

📝 Topic test — 8 questions

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