* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Section 7.2 Systems of Linear Equations in Three Variables
Survey
Document related concepts
Functional decomposition wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Law of large numbers wikipedia , lookup
Mathematical model wikipedia , lookup
Recurrence relation wikipedia , lookup
Analytical mechanics wikipedia , lookup
Elementary algebra wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary mathematics wikipedia , lookup
Signal-flow graph wikipedia , lookup
History of algebra wikipedia , lookup
Partial differential equation wikipedia , lookup
Transcript
Section 7.2 Systems of Linear Equations in Three Variables Overview • When solving systems of linear equations in two variables, we utilized the following techniques: 1. Substitution 2. Elimination 3. Graphing • In this section we will develop techniques that can be used for larger systems. Three-variable Systems • A system of linear equations in three variables is in the form Ax By Cz D Ex Fy Gz H Jx Ky Lz M • The solution to a three-variable system is an ordered triple (x,y,z). Example 1 • Determine if (2, -4, 4) is a solution to x 9 z 34 4 x 4 y 24 8 y 2 z 40 Remember This? • Solve the following system by elimination: 3x – 4y = 25 5x + y = 11 • We will use a similar strategy to solve a system that has three equations and three unknown variables. Steps In The Process 1. Pick any two equations and use elimination to cancel one variable (x, y, or z). 2. Pick another set of two equations and use elimination to cancel the same variable. 3. Now solve this new, smaller system for the remaining variables. 4. Take those values and substitute them into any of the three original equations to find the third and final variable. Example 2 5 x 10 y 3z 21 x 5y z 0 10 x y z 12 Example 3 3x y 3 y 2 z 1 x 4 y z 12 Example 4 • The sum of three numbers is 8. The sum of twice the first number, 3 times the second number, and 4 times the third number is 22. The difference between 5 times the first number and the second number is 11. Find the three numbers. Let’s Go Shopping! • On a recent trip to the convenience store, you picked up 4 gallons of milk, 5 bottles of water, and 6 snack-size bags of chips. Your total bill (before tax) was $26.40. If a bottle of water costs twice as much as a bag of chips, and a gallon of milk costs $1.80 more than a bottle of water, how much does each item cost? A Little Geometry Never Hurt Anyone… • Find the values of x, y, and z in the triangle below: y x 3x + 4 z 3x - 4