In short
A system of linear equations is consistent if it has at least one solution and inconsistent if it has no solution. Consistent systems split further: independent (exactly one solution — the lines cross at a single point) and dependent (infinitely many solutions — the two equations describe the same line). For a pair a_1 x + b_1 y = c_1 and a_2 x + b_2 y = c_2, compare the three ratios a_1/a_2, b_1/b_2, c_1/c_2: all three equal means dependent, the first two equal but the third different means inconsistent, and the first two different means independent.
The first time you read the word "consistent" in your NCERT Class 10 maths book, it sounds like one of those English words that math teachers use just to make a simple thing sound impressive. It is not. It is the cleanest one-word way to ask: do these equations agree with each other, or are they fighting?
The plain meaning
A system of equations is a list of conditions you want to satisfy at the same time. "Consistent" is the same word you would use in everyday life — your friend's story is consistent if all the parts fit together. A system is consistent if there exists at least one pair (x, y) that fits every equation at once.
If no such pair exists — if the equations contradict each other so badly that no (x, y) in the universe can satisfy both — the system is inconsistent. Why: geometrically, each linear equation in two variables is a line. A solution is a point that lies on every line at once, which means the lines must overlap somewhere. If they never overlap, there is nothing to find.
So:
- Consistent = the lines overlap somewhere (at one point, or everywhere along a shared line).
- Inconsistent = the lines never overlap.
That's the whole idea. The rest is just naming the two flavours of "consistent."
The trichotomy
Two straight lines in a plane can do exactly three things relative to each other. They can cross at one point, they can be parallel without ever touching, or they can be the same line drawn twice. That is it — there is no fourth option. This three-way split is called the trichotomy of linear systems.
| Type | Geometry | Number of solutions |
|---|---|---|
| Inconsistent | Parallel lines, different intercepts | 0 |
| Consistent independent | Lines cross at exactly one point | 1 |
| Consistent dependent | Lines coincide (the same line) | \infty |
The word independent here means the two equations carry independent information — each one tells you something the other doesn't, and together they pin down a single point. Dependent means the second equation is just a multiple of the first; it depends on the first and adds nothing new. A dependent system is consistent because every point on the shared line is a solution, so there are infinitely many.
The ratio test (NCERT's favourite)
You don't have to plot the lines to figure out which case you are in. There is a quick algebraic test using the coefficients. Given
compare the three ratios \dfrac{a_1}{a_2}, \dfrac{b_1}{b_2}, \dfrac{c_1}{c_2}:
- \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} → consistent dependent, infinite solutions. Why: every coefficient of the second equation is the same multiple of the first, so equation 2 is just equation 1 in disguise — the same line.
- \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2} → inconsistent, no solution. Why: the left sides are scalar multiples (parallel lines have the same slope) but the right sides are not in that ratio, so the lines are shifted apart and never meet.
- \dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2} → consistent independent, unique solution. Why: the slopes are different, so the lines are not parallel and must cross at exactly one point.
This is exactly the test the NCERT Class 10 chapter "Pair of Linear Equations in Two Variables" asks you to apply, and it shows up nearly every year in board exam MCQs and very-short-answer questions.
Three worked examples
No solution (inconsistent)
Form the ratios:
The first two are equal, the third is different. The system is inconsistent — no (x, y) satisfies both. Why: doubling the first equation gives 4x + 6y = 14, but the second equation says 4x + 6y = 11. The same expression cannot equal 14 and 11 at once. The lines are parallel.
Infinite solutions (consistent dependent)
Form the ratios:
All three are equal. The system is consistent dependent — every point on the line 2x + 3y = 7 is a solution, so there are infinitely many. Why: the second equation is just 2 \times the first. It contains zero new information. You have one equation, two unknowns, and a whole line of solutions.
Unique solution (consistent independent)
Form the first two ratios:
These are already different, so you don't even need to check the third. The system is consistent independent — exactly one solution exists. Why: the slopes of the two lines are different, so they cross at one and only one point. Solving: from the second equation y = 5x - 1, substitute into the first to get 2x + 3(5x - 1) = 7, so 17x = 10, giving x = 10/17 and y = 33/17.
Why the vocabulary matters
When you read a question that says "for what value of k is the system consistent?", the examiner is asking you to find the values of k for which a solution exists at all — that is, both the dependent and independent cases. "For what k is the system inconsistent?" asks you to force the lines to be parallel-but-not-identical. The ratio test handles both phrasings in a few lines.
The same words generalise. A system of three equations in three variables, or a 100 \times 100 system you might meet in an engineering course later, is still called consistent if at least one solution exists and inconsistent otherwise. The names you learn now will follow you all the way through linear algebra.