Elimination: remove one unknown at a time

The café puzzle as equations

Call the price of a coffee x and the price of a croissant y. The two orders say:

One subtraction, and x was gone. Every system is solved this way. We only need a rule for how much to subtract.

The same idea, step by step

Take Strang's first system from slide 1: x − 2y = 1 and 3x + 2y = 11.

• Equation 1 starts with 1x. That 1 is the pivot. It clears the x below.
• Equation 2 has 3x, so we need 3 copies of equation 1. That 3 is the multiplier, ℓ = 3 ÷ 1 = 3.
• Subtract 3 times equation 1 from equation 2. The x terms cancel.

Putting y = 1 back into equation 1 is called back substitution.

What it looks like

Draw the lines before and after the step. Line 1 stays where it is. Line 2 turns until it is horizontal: 8y = 8 is the line y = 1. The crossing point 31 never moves, because the step did not change the solution.

Why the answer does not change

Any xy that solves both equations also solves "equation 2 minus 3 times equation 1". And adding 3 times equation 1 back undoes the step. So the new system has exactly the same solutions, just easier to read.

Only the numbers matter

The letters x and y only hold the numbers in place. So we can drop them and keep a table. Each equation becomes one row: the number in front of x, the number in front of y, a bar, and the right side. This table is called the augmented matrix [A | b]. The step "row 2 − 3·row 1" acts on the whole row, the right side too.

A bigger example: 3 equations

Slide 1's bigger system works the same way. First use equation 1 to remove x from equations 2 and 3. Then use the new equation 2 to remove y from equation 3.

The answer x = 1, y = 2, z = 3 matches slide 1. Circuit simulators and bridge-design programs follow the same steps with thousands of equations. The computer just does the arithmetic.

Next: With three or more equations, the order of the steps matters. Slide 3 clears one column at a time, names the pivots, and climbs back up with back substitution.