Systems of Linear Equations Find Shared Constraints

A system asks for values satisfying several equations simultaneously. Each equation is a constraint; a solution lies in every constraint set.

Geometry first

Two linear equations in $x$ and $y$ represent two lines. They may intersect once, never intersect because they are parallel, or coincide and share infinitely many points.

Substitution

When one equation isolates a variable, substitute that expression into the other equation.

Elimination

Add equivalent multiples of equations so one variable cancels. For

$2x+3y=12,$ $5x-3y=9,$

addition gives $7x=21$, so $x=3$. Substitution gives $y=2$.

Multiplying an equation by a nonzero constant preserves its solution line, which is why elimination is valid.

Classify exceptional outcomes

If elimination produces $0=5$, the constraints are inconsistent: no solution. If it produces $0=0$, one equation was redundant: infinitely many solutions remain along a line.

Check your understanding

Classify the system $2x+4y=6$ and $x+2y=3$.

Show the reasoning

The first equation is exactly twice the second. They describe the same line, so there are infinitely many solutions.