* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Warm-UP
Debye–Hückel equation wikipedia , lookup
Schrödinger equation wikipedia , lookup
Two-body problem in general relativity wikipedia , lookup
BKL singularity wikipedia , lookup
Van der Waals equation wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Equations of motion wikipedia , lookup
Unification (computer science) wikipedia , lookup
Perturbation theory wikipedia , lookup
Equation of state wikipedia , lookup
Calculus of variations wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Itô diffusion wikipedia , lookup
Differential equation wikipedia , lookup
Schwarzschild geodesics wikipedia , lookup
Warm-Up • Tickets for a concert cost $10 each if you order them online, but you must pay an $8 service charge per order. The tickets are $12 each if you buy them at the door on the night of the concert. • • Write a system of equations to model the situation. Let y be the total cost. Let x be the number of tickets purchased. y 10 x 8 y 12 x 6.2: Solving Systems Algebraically Substitution method You can solve linear systems by solving one of the equations for one of VARIABLES the ______________. SUBSTITUTE the expression for that variable into the other Then, ______________ SUBSTITUTION METHOD equation. This is called the ________________________. SOLVED , When a system has at least one equation that can be easily __________ for a variable, the system can be solved efficiently using substitution. y 5x Example: What is the solution to the system x y 12 STEP 1: Substitute 5x for y in the second equation. STEP 2: Solve the equation for x. STEP 3: Replace x and solve for y in the first equation. STEP 4: Solution to the system: x (5 x) 12 6 x 12 x2 y 5(2) y 10 x=2 y = 10 Solving for a Variable and Using Substitution: 7 x 3 y 2 Example: What is the solution to the system 2 y x 6 2 y x 6 x 2y 6 STEP 1: Rewrite one of the equations so a variable is by itself. STEP 2: Substitute this value into the other equation. 7(2 y 6) 3 y 2 14 y 42 3 y 2 11 y 44 y4 STEP 3: Solve for the variable. STEP 4: Replace in the first equation. STEP 5: Solution to the system: x=2 y =4 7 x 3(4) 2 x2 Special Systems IDENTITY If you get an like 5 = 5 when solving a system this means there are an infinite number of solutions to the system. If you get a FALSE statement like 0 = -2 then there are no solutions to the system. Examples 1) How many solutions does each system have? y 4x 9 x 3 y 4 b.) a.) 6 y 2 x 8 y 4x 6 6 y 2(3 y 4) 8 6y 6y 8 8 88 Infinitely Many Solutions (4 x 9) 4 x 6 4x 9 4x 6 9 6 No Solution Homework: Sect. 6.2 p. 393-395 #’s 12-24 even