1
A solution satisfies every equation
An ordered pair (x, y) solves the system only if substituting it makes both equations true. Satisfying one of them is not partial credit; it is not a solution at all.
Lesson · Systems and Matrices · Systems
Systems of Two Linear Equations
A system asks which pairs satisfy several equations at once, and a solution has to satisfy every one of them. Two lines cross once, never, or everywhere — and those three cases are the whole classification.
one solution the lines cross once consistent, independent
no solution parallel and distinct inconsistent
infinitely many one line written twice consistent, dependent
Three cases, and nothing else
Core ideas
What this lesson establishes
A system is a question about several equations at once. Two things have to be settled before any method: what a solution is, and how many there can be.
2
There are exactly three possibilities
One, none, or infinitely many. Two lines cannot cross in exactly two points, because two distinct points already determine a single line — so sharing two points means sharing all of them.
3
A graph names the case; it does not give the answer
The lines 2x + y = 4 and x − y = 1 cross at (5/3, 2/3). A careful drawing puts the crossing near (1.7, 0.7), and no drawing distinguishes 5/3 from 1.7.
The material
Systems, solutions, and the three cases
One definition, then the complete list of what two lines can do.
Definition · System of two linear equations
A linear equation in x and y is one that can be written ax + by = c with a and b not both zero. A system is two or more such equations considered together, and a solution of the system is an ordered pair (x, y) that satisfies every equation in it.
The solution set is the set of all such pairs. Solving the system means describing that set, which is not always a matter of naming one pair.
"Considered together" is where the content is. On its own, 2x + y = 4 has infinitely many solutions — one for every value of x. The system asks for the pairs the two equations have in common, so the solution set of the system is the intersection of the two solution sets, never their union.
Lines, Functions and Graphs established that the graph of ax + by = c is a line. So the solution set of a two-equation system is the set of points lying on both lines, and the question "how many solutions" becomes the question "how do two lines meet".
Theorem · Two lines meet in nothing, one point, or a whole line
Two lines in the plane share no point, exactly one point, or every point of the line. No other count is possible — in particular no system of two linear equations has exactly two solutions.
Suppose the two lines share two distinct points P and Q. There is exactly one line through P and Q, so the two lines are the same line, and then every point of one lies on the other.
So if the lines share more than one point they share all of them. The remaining possibilities are one shared point and none.
□
The three cases carry names. A system with at least one solution is consistent; one with none is inconsistent. A consistent system whose equations describe the same line is dependent; one with exactly one solution is independent.
Telling the cases apart without drawing
Put both equations in slope-intercept form. Different slopes means the lines cross once. Equal slopes with different intercepts means parallel and distinct, so no solution. Equal slopes and equal intercepts means one line written two ways, so infinitely many.
2x + 3y = 6 y = −(2/3)x + 2 slope −2/3, intercept 2
4x + 6y = 7 y = −(2/3)x + 7/6 slope −2/3, intercept 7/6
same slope, different intercept → parallel, distinct → none
Vertical lines do not have slopes, so slope-intercept form is not always the right comparison. In standard form, ask whether one equation's two variable coefficients are a common multiple of the other's. If not, the lines cross once. If they are, the constants decide whether the equations are the same multiple or a contradiction. “Determinants and Cramer's Rule” packages that coefficient comparison into one number, the determinant.
Example · The same line, written twice
x − 2y = 4 and −3x + 6y = −12. The second equation is −3 times the first, so every pair satisfying one satisfies the other and the solution set is the whole line {(x, y) : x − 2y = 4}.
Infinitely many is not the same as "all pairs". (0, 0) fails both equations, and so does most of the plane.
Checking a candidate is not optional and takes ten seconds: substitute the pair into every original equation. A pair produced by an error usually satisfies the last equation you touched, which is exactly the one that will not catch it.
Interactive
Two lines, and where they meet
One solution, none, or infinitely many -- decided by rank, drawn by sampling
Method
Reading a system before solving it
Four questions, in this order. The first three often finish the job.
1
Write both equations the same way
Either both as ax + by = c or both as y = mx + b, with the terms in the same order. Two equations written in different shapes cannot be compared, and comparing them is the next three steps.
2
Compare the slopes
Different slopes means exactly one solution: stop classifying and go and find it. Equal slopes means the system is one of the two degenerate cases.
3
If the slopes agree, compare the intercepts
Different intercepts: parallel and distinct, no solution. Same intercept: the two equations describe one line, and the solution set is that line.
4
Solve, then substitute into both originals
The pair must be tested in every equation of the system, including the ones you stopped using early. A solution that satisfies one equation is not a solution.
Systems and their solutions
Score 0 / 0Choose the answer you can justify.
Classifying three systems
Each pair is put into slope-intercept form and read off. The count of solutions is settled before any solving happens, and in two of the three cases no solving is possible.
(a) 2x − y = 1 y = 2x − 1 slope 2
x + y = 5 y = −x + 5 slope −1 slopes differ
2x − 1 = −x + 5 → 3x = 6 → x = 2, y = 3
one solution (2, 3) check 4 − 3 = 1 ✓ 2 + 3 = 5 ✓
(b) 2x + 3y = 6 y = −(2/3)x + 2
4x + 6y = 7 y = −(2/3)x + 7/6
same slope, different intercept → no solution
(c) x − 2y = 4 y = (1/2)x − 2
−3x + 6y = −12 y = (1/2)x − 2
the same line → infinitely many: {(x, y) : x − 2y = 4}
System (b) has no solution, and that is a complete answer. Pushing on with algebra produces a false statement — doubling the first equation gives 4x + 6y = 12 against 4x + 6y = 7, so 0 = 5 — which is the signature “Solving by Substitution” and “Solving by Elimination” teach you to read rather than to fear.
System (c) is the one that gets reported wrongly. The answer is a line, and it has to be written as a line — either as the equation x − 2y = 4 or in the parameter form (2t + 4, t). "Infinitely many" on its own says how many and not which.
Faded rehearsal: classify x = 3 with 2x + y = 7. The first line is vertical, so do not force a slope comparison: substitute its fixed x into the other equation, finish the ordered pair, and check it in both originals. Then change the second equation to 2x = 8 and explain the different classification before doing any solving.
Common mistakes
1
Reporting x and stoppingThe solution of a two-variable system is a pair. x = 2 is half an answer, and the missing half is where the sign errors live.
2
Checking the answer in one equation onlyIf y was found by substituting into the second equation, that equation will check out however badly the first step went. The equation you have not used since the beginning is the one that catches the error.
3
Calling a dependent system's solution set everythingx − 2y = 4 and −3x + 6y = −12 are satisfied by infinitely many pairs, but (0, 0) is not one of them. The solution set is a line, not the plane, and it has to be described.
Check your understanding
✓
Finish when you can name the case before you solve.
From the coefficients alone you should be able to say whether a system has one solution, none, or infinitely many, and only then produce the pair and check it in both equations. Naming the case first is what turns an inconsistent system into a two-line answer instead of a page of algebra ending in 0 = 14.
Note: The three cases survive every enlargement of the problem. “Systems in Three Variables” has three variables and the same three answers; “Gaussian Elimination” reads them straight off a row-reduced matrix without any geometry at all. The determinant in “Determinants and Cramer's Rule” will compress the coefficient comparison into a single test for the first case.