Chapter 7 Linear Systems and Matrices Overview: 7.1 Solving Systems of Equations 7.2 Systems of Linear Equations in Two Variables 7.3 Multivariable Linear Systems
7.1 Solving Systems of Equations What You ll Learn: #121 - Use the method of substitution and the graphical method to solve systems of equations in two variables. #122 - Use systems of equations to model and solve real-life problems.
Solving Systems of Equations 2x + y = 5 equation 1 ቊ 3x 2y = 4 equation 2 We will learn different techniques for solving a system of equations. The Methods: 1. Substitution 2. Graphical 3. Elimination 4. Gaussian Elimination 5. Matrices 6. Cramer s Rule
The Method of Substitution 1. Solve for a variable (pick the easier one!) 2. Substitute the expression you found from step 1 into the other equation 3. Solve the equation obtained from step 2 4. Back-substitute the value found from step 3 into the expression obtained in step 1 to find the value of the other variable 5. Check to see if solution satisfies the system of equations
Solution to a System of Equations Remember: The solution to a system of equations is the x, y point of intersection between the two linear functions.
Example 1 Solve the system of equations. ቊ x + y = 4 x y = 2
Example 2 A total of $12,000 is invested in two funds paying 9% and 11% simple interest. The yearly interest is $1,180. How much is invested at each rate? Let x and y represent the amount of money in each fund. x + y = 12,000 ቊ 0.09x + 0.11y = 1180
Example 3 No Solution Solve the system of equations. ቊ x + y = 4 x 2 + y = 3
Example 4 Two-Solutions Solve the system of equations. ቊ x2 + 4x y = 7 2x y = 1
Example 5 - Graphically Solve the system of equations. y = ln x ቊ x + y = 1
Applications Total cost C of producing x units has two components: 1. Initial cost 2. Cost per unit When the total cost (initial cost + cost per unit) equals the total revenue, we refer to that as the break-even point.
Example 6 Break-Even Analysis A small business invests $10,000 in equipment to produce a new soft drink. Each bottle of the soft drink costs $0.65 to produce and is sold for $1.20. How many items must be sold before the business breaks even? ቊ C = cost per item (number of items) + (initial cost) R = (sale price per item) (number of items sold)
Example 7 State Populations From 1991 to 2001, the population of Idaho was increasing at a faster rate than the population of New Hampshire. Two models that approximate the populations P (inthousands) are P = 1019 + 28.5t ቊ P = 1080 + 15.7t where t represents the year, with t = 1 corresponding to 1991. 1. When should you expect the population of Idaho to exceed New Hampshire s? 2. Use the model to estimate the population of both states in 2006.
Geometry Examples The perimeter of a rectangle is 30 meters long and the length is twice as long as the width. Find the dimensions of the rectangle. What are the dimensions of a rectangular plot of land if its perimeter is 96 miles and its area is 380 square miles.
Homework Page 459 #1,7,8,12,28,29,63,65,68,71 #75,76,73abc
7.2 Systems of Linear EQ in Two Variables What You ll Learn: #123 - Use the method of elimination to solve systems of linear equations in two variables. #124 - Graphically interpret the number of solutions of a system of linear equations in two variables. #125 - Use systems of linear equations in two variables to model and solve real-life problems.
The Method of Elimination This method works by adding or subtracting the equations. Example: ቊ 3x + 5y = 7 3x 2y = 1 3x+5y=7 3x 2y= 1 3y = 6 The idea is to eliminate a variable.
Example 1 Solving the system of linear equations. ቊ 3x + 2y = 4 5x 2y = 8
The Method of Elimination 1. Obtain coefficients that differ only in sign. 2. Add the equations to eliminate one variable. Solve resulting equation. 3. Back-substitute the value obtained in Step 2 into either of the original equations to solve for the other variable. 4. Check your solution to see if it satisfies the system of equations.
Example 2 ቊ Solving the system of linear equations. 5x + 3y = 9 2x 4y = 14
Example 3 No Solution ቊ Solving the system of linear equations. x 2y = 3 2x + 4y = 1
Example 4 Infinitely Many Solutions ቊ Solving the system of linear equations. 2x y = 1 4x 2y = 2
Example 5 Decimals? Solving the system of linear equations. ቊ 0.02x 0.05y = 0.38 0.03x 0.04y = 1.04
Homework Page 469 #1-6,15,53
7.3 Multivariable Linear Systems What You ll Learn: #126 - Use back-substitution to solve linear systems in row-echelon form. Solving systems of equations in 3 variables #127 - Use Gaussian elimination to solve systems of linear equations.
Example 1 3 variables, (x, y, z) Solve the system of linear equations. x 2y + 3z = 9 ቐ x + 3y = 4 2x 5y + 5z = 17
Consistent vs Inconsistent A system of linear equations is called consistent if it has at least one solution A consistent system with only one solution is independent. A consistent system with many solutions is dependent. A system of linear equations is called inconsistent if it has no solution.
Example 2 Inconsistent Solve the system of linear equations. x 3y + z = 1 ቐ 2x y 2z = 2 x + 2y 3z = 1
Example 3 Consistent & Dependent Solve the system of linear equations. ቐ x + y 3z = 1 y z = 0 x + 2y = 1
Graphical Interpretation (a) One Solution (exactly one point of intersection) (b) Infinitely Many Solutions (all the points existing on the line)
Graphical Interpretation No common point of intersection.
Homework Page 483 #1,5,17-19
Partial Fraction Decomposition We can rewrite a rational expression as a sum of two simpler rational expressions: x+7 x 2 x 6 = 2 x 3 + 1 x+2
Example x+7 x 2 x 6
Example 1 x 2 1
More Money Problems A company has invested $10,000 in three separate accounts. The accounts had 7%, 8%, and 9.5% interest. The amount of money invested in the 7% account is half the amount invested in the 8% account. At the end of the year, the accumulated interest was $1,200. How much was invested in each account?
Homework Page 485 #54,58,60,85
Chapter 7 Review 7.1-7.3 Page 548 #1-6 *substitution #11,12 *using a graphing utility #13,14 *applications #17,19,23 *elimination #32,34 *applications #35,40 *three variables #45 *partial fraction