o Graphing Basics
-> A system of equations shares the same two variables.
-> Graph both related lines on one coordinate plane.
-> An intersection point satisfies both equations.
-> Plot intercepts or use slope from the y-intercept.
-> Example: y = 2x - 1 and y = -x + 5.
-> First line: (0, -1), then up 2 and right 1.
-> Second line: (0, 5), then down 1 and right 1.
-> The lines meet at (2, 3).
-> Check: 2(2) - 1 = 3 and -(2) + 5 = 3.
o Classification
-> One intersection means one solution.
-> The system is consistent and independent.
-> Identical lines mean infinitely many solutions.
-> The system is consistent and dependent.
-> Parallel distinct lines mean no solution.
-> The system is inconsistent.
-> Different slopes: one solution.
-> Same slope and same intercept: infinitely many.
-> Same slope and different intercepts: none.
o One Solution
-> Example: 3x + y = 10 and x - y = 2.
-> Solve each for y: y = -3x + 10 and y = x - 2.
-> Graph or equate: -3x + 10 = x - 2.
-> Add 3x: 10 = 4x - 2; add 2: 12 = 4x.
-> So x = 3 and y = 3 - 2 = 1.
-> The solution is (3, 1).
-> Check both: 3(3) + 1 = 10 and 3 - 1 = 2.
-> Always verify in both original equations, not just one.
o No Solution
-> Example: y = 4x + 1 and y = 4x - 3.
-> The slopes are equal but the intercepts differ.
-> The lines are parallel and never meet.
-> Equating gives 4x + 1 = 4x - 3.
-> Subtract 4x: 1 = -3, which is false.
-> A false statement signals no solution.
-> Do not invent an intersection from a rough sketch.
-> A true statement such as 0 = 0 signals dependent lines.
-> Then every point on the shared line solves the system.
o Technology
-> Solve each equation for y before entering it.
-> Example: 2.5y - 4x = 7.5 and y + 0.5x = 6.
-> First: y = 1.6x + 3.
-> Second: y = -0.5x + 6.
-> Enter both in a graphing tool and find the intersection.
-> The tool reports approximately (1.43, 5.29).
-> Check with exact fractions: x = 10/7.
-> Then y = -5/7 + 6 = 37/7.
-> The decimal display is rounded, not the exact solution.
o Equation as System
-> A single equation can be split at the equals sign.
-> Set each side equal to y to form a system.
-> Example: 5x - 2 = 13 - 3x.
-> System: y = 5x - 2 and y = 13 - 3x.
-> Equate: 5x - 2 = 13 - 3x.
-> Add 3x and 2: 8x = 15, so x = 15/8.
-> Check: 5(15/8) - 2 = 75/8 - 16/8 = 59/8.
-> Right side: 13 - 45/8 = 104/8 - 45/8 = 59/8.