Math101 · Independent Ontario learning library
Math101learn.math101.caSystems of Linear Equations
A system of linear equations asks for values that satisfy multiple linear relationships at the same time.
A solution to a system is an ordered pair that makes every equation in the system true.
One point, two conditions
Each linear equation represents a line and a set of infinitely many points. A system asks where those sets overlap. For two lines, an intersection point $(x,y)$ satisfies both equations simultaneously.
This is why solving each equation separately is not enough: the same values must work in both.
Three possible outcomes
Two lines can intersect once, never intersect, or be the same line.
| Geometry | Algebra | Solutions |
|---|---|---|
| different slopes | lines meet | one |
| same slope, different intercepts | parallel | none |
| equivalent equations | same line | infinitely many |
These outcomes appear in graphing, substitution, and elimination in different forms.
Checking a proposed solution
If either equation fails, the pair is not a system solution.
Choosing a method
Graphing shows the geometry and gives an estimate, but exact intersections may be hard to read. Substitution is efficient when one variable is already isolated or has coefficient $1$. Elimination is efficient when coefficients match or can be made opposites.
All valid methods must produce the same solution because they describe the same overlap.
A modelling example
Adult tickets cost $14$ dollars and student tickets cost $9$ dollars. A total of $120$ tickets earns $1380$ dollars. Let $a$ and $s$ be ticket counts:
The first equation models count; the second models revenue. Solving gives $a=60$ and $s=60$. Units and nonnegative whole-number restrictions belong to the interpretation.
Equivalent transformations
Replacing an equation by a nonzero multiple of itself does not change its line. Adding equations can remove a variable because every solution satisfies both. Substituting one equivalent expression for a variable also preserves the solution set.
These ideas justify the algorithms instead of making them collections of unexplained moves.
Special cases algebraically
If elimination produces $0=7$, the system is inconsistent and has no solution. If it produces $0=0$, the equations are dependent and describe the same line, giving infinitely many solutions.
Do not report $0$ as a variable value in either case; interpret the statement.
Common mistakes
Finding a point on only one line. Verify both equations.
Treating $x$ and $y$ answers independently. A solution is one ordered pair.
Ignoring context restrictions. Counts, lengths, and time may exclude algebraically possible values.
Calling $0=0$ the solution. It signals infinitely many ordered pairs, not one pair of zeros.
Quick self-check
- What does each equation represent?
- Which method makes the structure easiest?
- Did I find a complete ordered pair?
- Does substitution verify both original equations?
- Does the result make sense in context?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Which ordered pair satisfies both y = x + 2 and 2x + y = 10?
- 2x + (x + 2) = 10
- 3x = 8, so x = 8/3
- y = 8/3 + 2 = 14/3
End of lesson
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