5.1 Solving Systems of Linear Equations by Graphing

System of Linear Equations two linear eqns. considered at the same time Ex. x + y = 5 x – y = 1 solutions to systems of eqns. are all ordered pairs that are solns. to BOTH eqns. (both eqns. give a true stmt. when ordered pair is sub. in)

Ex. For the system: x + y = 5 x – y = 1 is (3, 2) a soln? x + y = 5 3 + 2 = 5 5 = 5 true x – y = 1 3 – 2 = 1 1 = 1 Since (3, 2) satisfies BOTH eqns, YES, it is a soln to the system (b) is (-1, 6) a soln? x + y = 5 -1 + 6 = 5 5 = 5 true x – y = 1 -1 – 6 = 1 -7 = 1 false Since (-1, 6) DOES NOT satisfy BOTH eqns, NO, it is NOT a soln to the system

Solving by Graphing Graph first eqn. Graph second eqn. on same set of axes Look for a point of intersection The point of intersection is the soln. If there is no point of intersection no solution If lines intersect everywhereinfinitely many solns. Check the soln. in BOTH eqns., if necessary

Ex. Solve the system by graphing: y = -2x + 1 x = -1 Graph y = -2x + 1 y-int: 1, m = -2/1 rise = -2, run = 1 Graph x = -1 (vert. line crossing x-axis at -1) Point of intersection is soln. (-1, 3) Check (-1, 3) in both eqns. y = -2x + 1 x = -1 3 = -2(-1) + 1 -1 = -1 3 = 2 + 1 true 3 = 3 true y 3 2 1 x -3 -2 -1 1 2 3 -1 -2 -3

Worksheet Notes: Inconsistent system: a system with no soln. (#2 on worksheet) Dependent eqns: eqns. that produce the same line (#3 on worksheet)

Summary One point of intersection solution: {(x, y)} Lines (distinct lines) slope  different No point of intersection inconsistent system No solution empty set Ø Lines (parallel) slope  same y-int  different Lines intersect everywhere dependent eqns. Infinite number of solutions solution: {(x, y)|eqn.} Lines (coincide - same line) slope  same y-int  same