One equation, two unknowns — a whole line of answers; two equations share just one.
Point 1 of 4 in this lesson: 10.4.1 Linear equations in two unknowns
Up to now an equation has had one unknown, and one number made it true. But the world keeps asking two questions at once: a hot dog and a soda cost $10 together, and the hot dog costs $2 more than the soda — how much is each? Two blanks, one sentence. In this lesson a new kind of equation is born — one with two unknowns — and you'll discover that a single such equation has not one answer but a whole line of them. Pair two equations together and the line of one crosses the line of the other at a single shared point. That crossing is the answer to both questions at once.
Keep our color habit as you read. The unknowns x and y are violet. Equation one — its line — is teal. Equation two — its line — is amber. And the one pair that satisfies both, the place where the lines meet, is the solution in red.
You already know a linear equation in one unknown, such as 2x + 3 = 11. It has a single letter, raised to the first power, and a single number answer. Now we let in a second letter.
A linear equation in two unknowns ties two letters together with a plus, a minus, and some numbers — and nothing fancier. The cleanest example is
x + y = 10.
"Two numbers that add to ten." The general shape is
ax + by = c,
where a, b, and c are numbers and — this is the one rule — a and b are not both zero (otherwise there's no unknown left to talk about). The word linear earns its name: each unknown appears to the first power only. No x², no xy, no x hiding under a square root.
Here is the move that makes two unknowns feel friendly: fix one, and the other is pinned. Take x + y = 10. If you decide that x = 3, the equation becomes 3 + y = 10 — a one-unknown equation you can already solve — so y = 7. Choose x = 8 instead and y = 2. You're free to pick x; once you do, y has no choice.
A linear equation in two unknowns, ax + by = c (with a, b not both zero), has both letters at the first power. Fixing one unknown turns it into an ordinary one-unknown equation that pins the other.
These are not linear: x² + y = 5 (a square), xy = 6 (two unknowns multiplied), 1x + y = 2 (an unknown in a denominator). Linear means flat: only first powers, only added and subtracted.
With one unknown, a "solution" was a single number. With two unknowns, a solution must say something about both letters at once — so it comes as a pair. We write it in order, x first, then y, inside parentheses:
(x, y) = (3, 7).
This is an ordered pair: the order matters. (3, 7) means x = 3 and y = 7 — not the other way around. A pair is a solution of an equation when plugging both numbers in makes the equation true together. Check (3, 7) in x + y = 10: 3 + 7 = 10 ✓. It works.
How many solutions are there? In Section 10.4.1 we saw we could pick any value for x and always find a matching y. So there are infinitely many solution pairs. Let's collect a few in a table for x + y = 10 (for each chosen x, take y = 10 − x):
| x | 0 | 2 | 4 | 6 | 10 |
|---|---|---|---|---|---|
| y | 10 | 8 | 6 | 4 | 0 |
Five of the infinitely many pairs that add to ten: (0,10), (2,8), (4,6), (6,4), (10,0).
Now plot those five pairs as points. Each pair (x, y) is an address on the plane: go right to x, then up to y. Watch what happens.
The dots line up. Connect them and you get a straight line — and that line is not just those five dots, it's all of them, the complete portrait of every solution. This is the heart of the lesson:
A solution of a two-unknown linear equation is an ordered pair (x, y) that makes the equation true. There are infinitely many, and they all lie on one straight line — the equation's graph. The line is the set of all solutions.
An ordered pair is a package deal. To test (5, 4) in x + y = 10, you must use both numbers: 5 + 4 = 9 ≠ 10, so (5, 4) is not a solution. Checking only one of the two coordinates tells you nothing.
Step x up and down. Watch y = 10 − x answer back and the point slide along the teal line — every stop is a solution. Then test a pair of your own to see if it lands on the line.
One equation gave a whole line of answers — too many to call "the answer." To pin things down we add a second condition that must hold at the same time. Back to the snack stand:
the two prices add to 10 and they differ by 2.
Two sentences, both about the same x and y, both true at once. In symbols:
The curly brace { is read "and." A system of linear equations in two unknowns is two (or more) linear equations in the same unknowns, required to be true together. Each equation, on its own, is still just a line of solutions:
So a system is really a question about two lines drawn on the same plane. The first line collects all the pairs satisfying equation one; the second collects all the pairs satisfying equation two. The interesting question is the one Section 10.4.4 answers: which pair, if any, is on both?
A system joins two linear equations in the same unknowns with a brace, meaning "both true at once." Geometrically it is two lines sharing one coordinate plane.
The solution of a system is the ordered pair that satisfies every equation in it — the one set of values that makes equation one true and equation two true. On the plane, a pair is on line one when it lies on the teal line, and on line two when it lies on the amber line. To be on both, it must sit where the two lines cross.
For our snack-stand system the crossing is (6, 4). We don't have to trust the picture — we can check it, the only proof that ever matters. Substitute x = 6, y = 4 into each equation:
| x + y = 6 + 4 = 10 | equation one ✓ |
| x − y = 6 − 4 = 2 | equation two ✓ |
Both hold, so (6, 4) really is the solution: the hot dog is $6, the soda is $4. (Try any other point on the teal line — say (7,3): it adds to 10 but 7 − 3 = 4 ≠ 2, so it fails equation two. Only the crossing passes both tests.)
Two straight lines on a plane usually cross once — one shared point, one solution. But two other things can happen, and they have plain meanings:
If two lines are parallel (same steepness, different heights), they never touch — the system has no solution. If the two equations are secretly the same line, every point on it works — infinitely many solutions. We'll treat these special cases fully later; for now, just know the crossing point isn't always one-and-only, even though it almost always is.
The solution of a system is the ordered pair on both lines — the point where they cross. Always confirm it by substituting back into both equations. Two lines may instead be parallel (no solution) or identical (infinitely many).
Set the right-hand numbers of x + y = c₁ and x − y = c₂. The widget plots both lines, finds where they cross, and prints the solution pair. When the crossing has whole-number coordinates, the red dot lands right on a grid corner.
Pick a pair (x, y) and drop it onto the snack-stand system x + y = 10, x − y = 2. The tester checks each equation separately, then tells you whether the point is the shared solution. Only one pair turns both checks green.
A linear equation in two unknowns, ax + by = c, keeps both letters at the first power; fixing one pins the other, so its solutions are ordered pairs (x, y) — infinitely many of them, lined up on a single straight line that is the picture of every solution. Stack two such equations into a system with a brace and you've drawn two lines; the solution of the system is the pair that satisfies both — the point where the lines cross — which you always confirm by substituting back into each equation. Most systems cross once; a few are parallel (no solution) or the same line (infinitely many).
Reading a crossing point off a graph is quick but only as exact as your eyes. In Lesson 10.5 · Solving Two-Unknown Systems: Elimination you'll find the solution with arithmetic instead of a ruler — add or subtract the two equations to make one unknown vanish, then back-solve for the other. For x + y = 10 and x − y = 2, adding them erases y in a single line.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson lays the conceptual foundation for U.S. Common Core grade 8. 8.F.A — a linear equation in two variables graphs as a straight line — is the backbone of Sections 10.4.1–10.4.2: students see that the line is the complete set of ordered-pair solutions. 8.EE.C.8a says the solution of a system corresponds to the point where the two lines intersect, which is exactly the move in Sections 10.4.3–10.4.4: a brace means "both true," and "both" geometrically means "on both lines," i.e. the crossing. 8.EE.C.8b (solving systems) is introduced here by graphing and checking, and is carried forward to elimination in Lesson 10.5.
The #1 misconception: treating an ordered pair like a one-unknown solution — checking only x (or only y), or accepting a pair that satisfies one equation of a system as the system's answer. The antidote: insist on substituting both numbers into every equation, every time. A pair is a package; "the solution of a system" must turn both checks green. Tie it to the picture — passing one equation puts you on one line; the system's answer must sit on both lines at once, and that is only the crossing point.