Two simultaneous equations are inconsistent when no pair of values can satisfy both. On a graph, that means two parallel lines that never meet. If the two equations are really the same line, there are infinitely many solutions instead.
This lesson completes simultaneous relationships. It shows you the two “other” outcomes, so that when the working does not produce neat values you know what has happened.
Three possible outcomes
| Outcome | What the graphs look like | What your working shows |
|---|---|---|
| One solution | Lines cross at one point | Both letters get a value |
| No solution | Parallel lines | A false line such as 0 = 3 |
| Infinitely many | Same line | A true line such as 0 = 0 |
A quick check: lines with the same gradient but different intercepts are parallel. Lines with the same gradient and the same intercept are identical.
The method, step by step
- Line up the equations and compare the coefficients of x and y.
- Multiply one equation so the x-coefficients (or y-coefficients) match.
- Subtract to eliminate the letter.
- If both letters vanish, look at what is left. A false statement means no solution. A true statement means infinitely many.
- Write the conclusion in words.
Worked example
Solve: 2x + 3y = 6 (1) and 4x + 6y = 15 (2)
Step 1, match the x-coefficients. (1) × 2 gives 4x + 6y = 12.
Step 2, subtract this from (2): (4x + 6y) − (4x + 6y) = 15 − 12, which gives 0 = 3.
Step 3, read the result. Both letters have disappeared and the statement 0 = 3 is false.
Conclusion: the equations are inconsistent. The left-hand side of (2) is exactly twice the left-hand side of (1), but the right-hand side is 15 instead of 12. The lines are parallel, so there is no solution.
A contrasting pair: x + 2y = 4 and 3x + 6y = 12. Multiply the first by 3 to get 3x + 6y = 12, which is identical to the second. Subtracting gives 0 = 0, so there are infinitely many solutions.
The mistake to watch for
The usual slip is to treat 0 = 3 as if it were an equation to solve.
Mistaken working: “0 = 3, so x = 0 and y = 3.”
The student read the numbers as if they were values for the letters. But 0 = 3 has no letters left, and it is simply false.
The correction is to stop and ask what the equation says about the situation. A false statement cannot be fixed by finding x and y. Write “no solution, the lines are parallel” and, if the question asks, justify it by comparing the gradients or by showing the false line.
Check yourself
Try these on paper, then open each answer.
1. How many solutions do 3x − y = 5 and 6x − 2y = 7 have?
Show answer
Double (1): 6x − 2y = 10. Compare with (2): 6x − 2y = 7. The left sides match but 10 ≠ 7, so subtracting gives 0 = 3, which is false.
No solution. The lines are parallel.
2. How many solutions do x + 2y = 4 and 3x + 6y = 12 have?
Show answer
Triple (1): 3x + 6y = 12, which is exactly (2). Subtracting gives 0 = 0, which is always true.
Infinitely many solutions. Both equations describe the same line.
3. Which of these pairs has exactly one solution? (a) y = 3x + 1 and y = 3x − 4 (b) 2x + y = 5 and x − y = 1. Find it.
Show answer
(a) Both gradients are 3 and the intercepts differ, so the lines are parallel with no solution. (b) Add the equations: 3x = 6, so x = 2. Then 2 − y = 1 gives y = 1. Check (1): 4 + 1 = 5, which matches.
Pair (b), with x = 2, y = 1.
Where this leads next
You now have all the outcomes for two linear equations. Test them together in the simultaneous relationships practice set, and use the non-calculator working trainer to check your arithmetic as you go. If you want to revisit the elimination routine, go back to solving by elimination.
Students who panic when the expected numbers do not appear often just need to have seen this pattern once with someone explaining it. That kind of close attention to working is a central part of online one-to-one Mathematics tuition.