7.3 Part 1 Multi-Variable Systems and Applications- 2 Days

7.3 (Day 1-2 of 4) Day 1: Multi-Variable Systems 7.3 Day 2: Applications

Example 1 This is an example of a triangular system of linear equations. Use back-substitution to solve this triangular system. 2x – y + 5z = 16 y + 2z = 2 z = 2

Example 2 Solve the system. x + y + z = 6 2x – y + z = 3 3x - z = 0

Solution (1, 2, 3), check please.

Example 3 Solve the system of equations: x – y + z = 4 x + 3y - 2z = -3 3x + 2y + z =5

Solution (2, -1, 1), check please.

Example 4 Solve the system of linear equations. x + y – 3z = –1 y – z = 0 –x + 2y = 1

Solution Example 4 So any ordered triple solution of this system of equations will be in this form: (2z – 1,z,z) Now find 2 solutions of this form. Don’t forget to check these solutions!

Non-square Systems In a non-square system of equations, the number of equations differs from the number of variables. Non-square systems will not have one unique solution, but can have infinitely many solutions.

Example 5 Solve the system of linear equations. x – 2 y +z = 2 2x – y – z = 1

Solution Example 5 So any ordered triple solution of this system of equations will be in this form: (z, z – 1, z) Now find 2 solutions of this form. Don’t forget to check these solutions!

Example 6- You Try Solve the system of linear equations. 5x–12y+7z =16 3x–7y+4z = 9

Infinite solutions that satisfy a line

Infinite solutions that satisfy a plane

Example 7 Solve the system of linear equations. x – 3 y + z = 1 2x – y – 2z = 2 x + 2y – 3z = – 1

Solution Example 7 Because you obtain a false statement, the system is inconsistent and has no solution.

No Solution

Application: You spent $20 on 3 types of binders. Red binders cost $2 each, blue binders cost $4 each and green binders cost $5 each. How many of each type of binder did you buy if you bought 6 binders total and the same number of red binders as blue and green combined?

Homework 7.3 Day 1: Pg. 483: 5-9 odds, 13,15,17,19, 27,33,35

7.3 Day 2: Applications 2016

Precalculus 7-3: Applications Find the equation of the circle: that passes through the following points:

Precalculus 7-3: Applications Vertical Motion: The height (s) at time t of an object with constant acceleration (a) is given by, where is initial velocity and is initial height. Find a,, and, and the vertical motion model if:

You try: Find the equation of the parabola: that passes through the following points by writing and solving a system of equations:

You try: Find a quadratic: whose graph passes through the following points by writing and solving a system of equations:

A small corporation borrowed $775,000 to expand its software line. Some of the money was borrowed at 8%, some at 9% and some at 10%. How much was borrowed at each rate if the annual interest was $67,000 and the amount borrowed at 8% was four times the amount borrowed at 10%?

You try: An investor with a portfolio of $325,000 has invested in certificates of deposit, bonds, and stocks. The cds pay 3% annually, the bonds pay 5.5% annually, and stocks are expected to return 7% annually. The investor wants a combined annual return of 18,250. How much was invested in each fund if twice as much was invested in stocks as cds.

You try: Nancy wants to purchase muffins for her book club. At Muffin Central, apple muffins cost $1.50 each, blueberry muffins cost $2.25 each and cranberry muffins cost $2.50 each. Nancy spent $71 on 33 muffins and she bought twice as many blueberry as the other types combined. How many of each type did she buy?

A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from 3 solutions whose concentrations are 10%, 20%, and 50%. How many liters of each solution should the chemist use so that as little as possible of the 50% solution is used?

7.3 Day 2 Homework Pg. 485: odd, 86, 87, 88, 92