Math 166, Fall 2011,cBenjamin Aurispa4.3-4.4 Systems of EquationsA linear equation in 2 variables is an equation of the form ax + by = c.A linear equation in 3 variables is an equation of the form ax + by + cz = d.To solve a system of equations means to find the set of points that satisfy EVERY equation in the system.A solution to a system of linear equations with 2 variables will be an ordered pair (e, f).A solution to a system of linear equations with 3 variables will be an ordered triple (e, f, g).The same ideas are true for 4 variables, etc.We can use algebraic methods to solve systems of equations, such as substitution and elimination.Solve the following system of equations algebraically.x − 2y = −83x + y = −3We would say this system of equations has 1 unique solution.When solving any system of linear equations, there are ONLY 3 possibilities for the number of solutions:1. Exactly One Solution (called a unique solution).2. No Solution (This is called an inconsistent system.)3. Infinitely Many Solutions (This is called a dependent system.)Solving a system of linear equations means finding the points of intersection. Think about two lines: Theywill either intersect at one point (unique solution), at no points (parallel lines), or at infinitely many points(same line).Instead of using algebra, we will learn a way to solve systems of equations using matrices and the calculator.A matrix is just a rectangular array of numbers.We can represent a system of equations with an augmented matrix. An augmented matrix is a short-handway of representing a system without having to write the variables.When writing an augmented matrix from a system of equations, line up all the variables on one side of theequal sign with the constants on the other side. Then, form a matrix with the coefficients and constants inthe equations.1Math 166, Fall 2011,cBenjamin AurispaExamples: Represent the following systems of equations with an augmented matrix.−3x + 2y − 6z = 14 2x − 4y = 94x − 5y + 2z = −3 −2y − z = 07x + 9y + 3z = 14 x + y + 4 = 2z3y + z = 5xIf a matrix has m rows and n columns, we say the size of the matrix is m x n. What are the sizes of theabove matrices?In order to solve a system, we need to “reduce” the matrix to a form where we can identify the solution.This form is called “reduced row echelon form.” It is equivalent to the original system, just simplified.The calculator command rref will reduce the matrix for you without having to do algebra.To enter a matrix into your calculator and find the row-reduced form:• Press 2nd x−1. (The TI-83 has a separate MATRX button.)• Cursor over to EDIT and select where you want to store your matrix. Enter the size of the matrixand then enter in the matrix values.• Go back to your home screen. Press MATRX and then cursor right to MATH. Scroll down and selectB:rref.• Select the matrix you want to reduce by pressing MATRX and then choosing your matrix under theNAMES column. Then close your parentheses and press enter. Note: The rref command only workswhen the number of rows is less than or equal to the number of columns.Once you have the reduced form of the matrix, you can determine the solution by rewriting each row backas an equation.Solve the first system from above:−3 2 −6 144 −5 2 −37 9 3 14Solve the second system from above:2 −4 0 90 −2 −1 01 1 −2 −4−5 3 1 02Math 166, Fall 2011,cBenjamin AurispaSolve the following systems:x + 2y = 2−3x + 3y = −17x − 13y = −1−x + 2y + 3z = 142x − y + 2z = 2−x + 5y + 11z = 44When a system has infinitely many solutions, we parameterize the solution (write the solution in parametricform):A particular solution is a specific solution to the system. What are some particular solutions for this system?Examples: The following reduced matrices represent systems of equations with variables x, y, z, and, ifnecessary, u. Determine the solutions to these systems.1 0 0 0 30 1 0 0 −40 0 1 2 30 0 0 0 11 0 0 0 60 1 0 0 −90 0 1 0 80 0 0 1 00 0 0 0 03Math 166, Fall 2011,cBenjamin Aurispa1 0 2 0 −50 1 5 0 00 0 0 1 20 0 0 0 0"1 0 2 −1 60 1 −3 4 1#"1 0 0 0 00 0 1 2 3#Systems of equations are used to model problems. For the following examples, set up a system of equationsto solve the problem and solve. ALWAYS DEFINE YOUR VARIABLES when setting up a system ofequations.Ben has a total of $20,000 invested in two municipal bonds that have yields of 7% and 11% simple interestper year, respectively. If the total interest he receives from the bonds in 1 year is $1880, how much does hehave invested in each bond?4Math 166, Fall 2011,cBenjamin AurispaA theater has a seating capacity of 900 and charges $2.50 for children, $4 for students, and $5.50 foradults. At a certain screening with full attendance, there were half as many adults as children and studentscombined. The total money brought in was $3825. How many children, students, and adults attended theshow?A sporting goods stores sells footballs, basketballs, and volleyballs. A football costs $35, a basketball costs$25, and a volleyball costs $15. On a given day, the store sold 5 times as many footballs as volleyballs. Theybrought in a total of $3750 that day, and the money made from basketballs alone was 4 times the moneymade from volleyballs alone. How many footballs, basketballs, and volleyballs were sold? Just set up theproblem.(Problem #40, Section 4.3) A furniture company makes loungers, chairs, and footstools out of wood, fabric,and stuffing. The number of units of each of these materials needed for each of the products is given in thetable below. How many of each product can be made if there are 1110 units of wood, 880 units of fabric,and 660 units of stuffing available? Just set up the problem.Wood Fabric StuffingLounger 40 40 20Chair 30 20 20Footstool 20 10 105Math 166, Fall 2011,cBenjamin AurispaIf a system of equations arising from a word problem has a parametric solution, sometimes there reallyaren’t infinitely many solutions, since only some of them will make sense in the problem.For example, if x represents the number of children that were at a movie, we would know that x ≥ 0 andthat x must be a whole number [an integer].(Problem #49, Section 4.4) A person has 36 coins made up of nickels, dimes, and quarters. If the totalvalue of the