Two Truths at Once
A system's solution is the one pair (x, y) that makes both equations true at once — the point where the two lines cross. · 11 min
So far, one equation at a time. But questions arrive in pairs more often than you might think. Two phone plans — when do they cost the same? Adult and child tickets — nine seats sold, 51 dollars taken — how many of each? Each condition is one equation, and the answer must honor both at once. A pair of linear equations taken together is called a system, and this folio is about what its solution actually means.
Guess before you learn
y = x + 1 and y = −x + 5 are drawn on the same axes. How many points land on both lines?
One line climbs and the other falls, so they cross exactly once — at (2, 3). Check it in both: 3 = 2 + 1 and 3 = −2 + 5. If you guessed two or many, hold the thought: straight lines are rigid, and this folio shows why one crossing is the usual case — and what the two exceptions look like.
9–12
3–5
Two clues about two mystery numbers: they add to 10, and the first is 2 bigger than the second. Try pairs. 5 and 5 add to 10 but break the second clue. 6 and 4 pass both — and no other pair does.
One clue alone leaves many possibilities. The second clue narrows them to a single pair. That is a system doing its job: two conditions, one answer.
6–8
A system of equations is two equations about the same x and y. A solution is one pair (x, y) that makes both equations true at the same time — not one pair each. Graphs make this visible: each equation draws the line of all pairs that satisfy it, so a pair satisfying both must sit on both lines. The crossing point is the solution.
To test a candidate pair, substitute it into both equations. Passing one is not enough — a solution answers to the whole system.
9–12
Each equation's graph is the set of every pair satisfying it, so the system's solution set is the intersection of the two lines — and geometry allows exactly three outcomes. Lines with different slopes must cross once: one solution. Lines with equal slopes and different intercepts are parallel: they never meet, so there is no solution. And if both equations draw the same line, every point of it works: infinitely many solutions.
So the count can be read before any solving: put both equations in y = mx + b and compare. Different m: one solution. Same m, different b: none. Same m, same b: infinitely many.
K–2
Ana hides a sticker and gives two clues. Clue one: it is on the top shelf. Clue two: it is in a red box. Only one spot makes both clues true.
A system is two clues at once. The answer must make clue one true and clue two true — both, together.
Undergrad
Write the system as Ax = c with A a 2 × 2 matrix. A unique solution exists precisely when det A ≠ 0 — which, for slope-intercept rows, is exactly the condition m₁ ≠ m₂. When det A = 0 the left-hand sides are proportional, and the system is either inconsistent (parallel lines) or redundant (coincident), according to whether the right-hand sides respect the same proportion.
The trichotomy is algebraic, not merely pictorial: one point, the empty set, or a whole line — never exactly two points, since two shared points would force the lines to coincide.
Postgrad
Rouché–Capelli in miniature: the solution set is empty or an affine coset of ker A, so its dimension here is 0 or 1 — a point, the empty set, or a line. Rank sorts the cases: rank A = 2 gives uniqueness; rank A = 1 splits on whether the augmented matrix keeps rank 1.
Projectively the trichotomy collapses: two distinct lines in the projective plane always meet in exactly one point, parallels meeting on the line at infinity. The affine 'no solution' is a solution lying outside the chart — a first look at why compactification simplifies intersection counting.
system of equations
Two or more equations about the same unknowns, taken together. A solution must satisfy every equation at once.
How many solutions can a system have? Read the lines. Different slopes: the lines must cross exactly once — one solution. Same slope, different intercepts: parallel lines, the same vertical gap apart forever, never meeting — no solution. Same slope, same intercept: the two equations draw one line twice, and every point of it works — infinitely many solutions. Nothing else can happen, because two distinct straight lines cannot share two points; if they did, they would be the same line.
How many solutions? y = 2x + 3 and y = 2x − 1 — the steps fade as you master them
m₁ = 2 and m₂ = 2 — equal
b₁ = 3 and b₂ = −1 — different
parallel lines — no solution
Why is this true?
Why can two different lines never cross at two separate points?
Two points determine exactly one straight line. If two lines both passed through the same two points, they would be that one line — so distinct lines share at most one point.
You can now say precisely what a system asks, and count its answers before solving anything. What you cannot yet do is name the crossing exactly when the graph is too coarse to read. The next folio hands you two algebraic tools — substitution and elimination — that find it to the digit.
Note
The gate above quietly reused folio 7: setting two expressions for y equal produces an equation with x on both sides. Old tools carry forward.
Practice — new ink and old, interleaved
1.Find the slope of the line through (−1, 4) and (3, −4).
2.Order the checks for counting a system's solutions.
- If the slopes differ, stop: exactly one solution
- Write both equations in y = mx + b
- If the slopes match, compare intercepts: different means none, same means infinitely many
3.A classmate says (5, 4) solves a system because it works in the second equation. In one sentence: what must still be checked?
4.Two distinct parallel lines form a system. How many solutions does it have?
5.6x + 1 = 6x + 1. How many solutions?
6.Which pair solves the system y = 2x − 1 and y = x + 1?
7.Which pair of points lies on a horizontal line?