📐 Linear Equations & Systems

• Elimination: add or subtract the two equations so one letter disappears. From $a+b=8$ and $a-b=6$, adding gives $2a=14$, so $a=7$.

• Substitution: if a point lies on a line, its coordinates satisfy the equation — just plug them in.

• Word problems: name the unknown, write one equation per sentence, then solve. In age problems the gap between two people never changes.

For two lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$: infinitely many solutions when $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$, and no solution when $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$.