Solving Systems of Equations

Method 1: Substitution

Example: $\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}$

Substitute $y = 2x + 1$ into second equation: $3x + (2x + 1) = 16$ $5x = 15$ $x = 3, \quad y = 7$

Method 2: Elimination

Example: $\begin{cases} 2x + 3y = 12 \\ 5x - 3y = 9 \end{cases}$

Add the equations: $7x = 21$ $x = 3, \quad y = 2$

Solution Types

• One solution: Lines intersect
• No solution: Parallel lines
• Infinitely many: Same line