Two pictures of the same equations.
Strang's first system: x − 2y = 1 and 3x + 2y = 11. You can look at it by rows or by columns. Both pictures give the same answer, x = 3 and y = 1.
Row picture: two clues
A navigator takes two compass bearings. Each bearing says "the ship is somewhere on this line". The ship is where the two lines cross.
Each equation works the same way. All points with x − 2y = 1 form one line. All points with 3x + 2y = 11 form another. The solution is the point on both: (3, 1).
Column picture: a recipe
Now read the system as a mix of the columns, like the basket on slide 2:
The question changes from "where do the lines cross?" to "how much of each column makes b?". Same x, different picture.
When it goes wrong
- Parallel lines: the clues never agree, so there is no solution. In the column picture, the columns are dependent and b is off their line.
- The same line twice: the second clue tells you nothing new. Infinitely many solutions, and b sits on the line of the columns.
Why Strang prefers columns
With three unknowns, each clue becomes a plane in space, and three planes crossing is hard to draw. The column picture stays simple: three arrows, one target.
The row picture is still how you solve by hand. The next lesson, elimination, works on rows.
Worked examples
Example 1 · One crossing
x + y = 3 and x − y = 1.
Example 2 · The same line twice
x + 2y = 4 and 2x + 4y = 8.
Infinitely many solutions, for example (4, 0) and (0, 2).
Problems
Try each one before you open the solution.
Solve 2x + y = 7 and x − y = 2, then check with the column picture.
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Adding the equations: 3x = 9, so x = 3 and y = 1.
Columns: 3·(2, 1) + 1·(1, −1) = (7, 2) ✓
What happens with x + y = 1 and x + y = 3? Describe both pictures.
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Rows: two parallel lines, no crossing, no solution.
Columns: both columns are (1, 1). Their combinations stay on the line through (1, 1), and b = (1, 3) is off it.
Write 3x − y = 5, x + 2y = 4 as a column picture and solve it.
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x·(3, 1) + y·(−1, 2) = (5, 4). Solution x = 2, y = 1: 2·(3, 1) + 1·(−1, 2) = (5, 4) ✓