Chapter 7 Solving systems of equations substitution (7-1) elimination (7-1) graphically (7-1) augmented matrix (7-3) inverse matrix (7-3) Cramer’s Rule (not in this book)

What’s in the Rest of Chapter 7 Section 7-2: matrices (mult., determinants, inverses) Section 7-4: partial fraction decomposition (uses systems of equation) Section 7-5: systems of inequalities and linear programming

Section 7-1 Solve by substitution Example: solve the system x + 2y = – 7 2x – 3y = 0

Section 7-1 Solve by substitution Example: solve the system x + 2y = – 7 2x – 3y = 0 change the first equation to x = (–2y – 7) substitute into the other equation and solve 2(–2y – 7) – 3y = 0 – 4y – 14 – 3y = 0 – 7y = 14 y = – 2

Section 7-1 Solve by substitution Example: solve the system x + 2y = – 7 2x – 3y = 0 change the first equation to x = (–2y – 7) substitute into the other equation and solve 2(–2y – 7) – 3y = 0 – 4y – 14 – 3y = 0 – 7y = 14 y = – 2 substitute answer into expression: -2(-2) – 7 x = – 3 so answer is (–3, –2)

Example: solve by substitution: x = 0, x = 3, and x = -3 plug into y = 3x to get their y-values (0, 0), (3, 9), and (-3, -9) are the solutions

Solve by elimination create opposites by multiplying one or both equations by a number add the two equations together to eliminate one of the variables solve the remaining equation plug 1 st answer back into either original equation to find the 2 nd answer

Solve by elimination: 3x + 7y = 15 5x + 2y = – 4 3x + 7y = 15 multiply by 2 5x + 2y = - 4 multiply by –7 6x + 14y = x – 14y = x = 58 add the two equations x = -2 solve, & plug in to find y y = 3 so the answer is (-2, 3)

Solve graphically solve both equations for y graph both equations and adjust the window find all intersection points Example: find all solutions to the system