2. Systems of Two Equations

A system is two equations that share the same two unknowns, for example:
\[ \begin{cases} x + y = 10 \\ x - y = 4 \end{cases} \]

Method A - Elimination: Add the two equations so one variable cancels.
\[ (x + y) + (x - y) = 10 + 4 \;\Rightarrow\; 2x = 14 \;\Rightarrow\; x = 7 \]
Then substitute \(x = 7\) into \(x + y = 10\): \(7 + y = 10 \Rightarrow y = 3\).

Solution: \(x = 7,\; y = 3\).

Method B - Substitution: Solve one equation for a variable, then plug it into the other. From \(x - y = 4\) we get \(x = y + 4\). Substitute:
\[ (y + 4) + y = 10 \;\Rightarrow\; 2y + 4 = 10 \;\Rightarrow\; y = 3,\; x = 7 \]

Both methods give the same answer. Use elimination when a variable lines up nicely; use substitution when one equation is already solved for a variable.