Systems of Linear Equations

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.

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.

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?