Topic 1 of 8
Systems and Gaussian elimination
Step 19 · Linear algebra topics
Write linear systems as augmented matrices, solve them by row reduction, and tell whether a system has one solution, none, or infinitely many.
Linear systems
A linear system is a set of equations in which each unknown appears only to the first power. With two unknowns each equation is a line, and a solution is a point on all the lines.
Example: Two lines
Solve x + y = 5 and 2x − y = 1.
- Add the equations to remove y: 3x = 6, so x = 2.
- Then y = 5 − 2 = 3.
Answer(2, 3), where the lines cross
Record only the numbers in an augmented matrix: one row per equation, one column per unknown, and a last column for the right-hand sides.
Row reduction
Three row operations never change the solutions: swap two rows, multiply a row by a number that is not 0, and add a multiple of one row to another. Use them to reach echelon form: a staircase of leading entries, called pivots.
Example: Three equations
Solve x + y + z = 6, 2x + y − z = 1 and x − y + 2z = 5.
- R2 − 2R1 gives −y − 3z = −11, and R3 − R1 gives −2y + z = −1.
- The new R3 minus twice the new R2 gives 7z = 21, so z = 3.
- Back-substitute: −y − 9 = −11 gives y = 2, and then x = 6 − 2 − 3 = 1.
Answer(1, 2, 3)
Solution sets
A system has no solution, exactly one, or infinitely many. Row reduction shows which: a row reading 0 = 5 means none; a pivot in every unknown's column means exactly one; a column without a pivot gives a free variable and infinitely many.
Example: No solution
Solve x + 2y = 3 and 2x + 4y = 7.
- R2 − 2R1 gives 0 = 1.
- That can never hold.
AnswerNo solution: the lines are parallel
Example: Infinitely many
Solve x + y + z = 3 and x + 2y + 3z = 5.
- R2 − R1 gives y + 2z = 2.
- z has no pivot, so it is free: call it t.
- Then y = 2 − 2t and x = 3 − y − z, which is 1 + t.
Answer(1 + t, 2 − 2t, t) for any t: a whole line of solutions
| After row reduction | Solutions |
|---|---|
| A row 0 = c with c not 0 | None |
| A pivot in every unknown's column | Exactly one |
| A column with no pivot, and no 0 = c row | Infinitely many |
Try it yourself
Work each one on paper first, then open the answer to check.
Solve x + y = 7 and x − y = 1.
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x = 4, y = 3
Write the augmented matrix of 2x − y = 3 and x + 4y = 6.
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Does the system x + y = 2, 2x + 2y = 5 have a solution?
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No: row reduction gives 0 = 1
An echelon form has a free variable and no 0 = c row. How many solutions are there?
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Infinitely many
Solve x + 2y = 4 and 3x + 5y = 11.
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x = 2, y = 1
Can three equations in two unknowns have exactly one solution?
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Yes, if all three lines pass through one point