Linear Equations & Graphs

For a system of equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:

• Unique Solution: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ (Lines intersect)
• No Solution (Parallel Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
• Infinite Solutions (Coincident Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

• $x = a$: A vertical line parallel to the Y-axis.
• $y = b$: A horizontal line parallel to the X-axis.
• $ax + by + c = 0$: A straight line. To plot, find X-intercept (put $y=0$) and Y-intercept (put $x=0$).