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