4.3-4.4: Systems of Linear Equations
4.3-4.4: Systems of Linear Equations A 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, 5 variables, etc. We can use algebraic methods to solve systems of equations, such as substitution and elimination. Given a system of linear equations, one and only one of the following may occur: 1. The system has a unique solution. Example 1. Find all solutions to the following system of equations, whenever they exist. 2x − 4y = 5 3x + 2y = 6 1 2. The system has infinitely many solutions. Such a system is said to be . Example 2. Find all solutions to the following system of equations, whenever they exist. 5x − 6y = 8 10x − 12y = 16 3. The system has no solutions. Such a system is said to be . 2 Example 3. Find all solutions to the following system of equations, whenever they exist. −10x + 15y = −3 4x − 6y = −3 Instead of using algebra, we will learn a way to solve systems of equations using matrices and the calculator. A is a rectangular array of numbers. We can represent a system of equations with an . An augmented matrix is a short-hand way 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 the equal sign with the constants on the other side. Then, form a matrix with the coefficients and constants in the equations. Example 4. Write the following system of equations as an augmented matrix. 2x − 2y = −1 −x + 5y + z = 3 3x + 4y + 5z = 2 Example 5. Write the following system of equations as an augmented matrix. 2x − 4y = 9 −2y − z = 0 x + y + 4 = 2z 3y + z = 5x 3 Example 6. Write the following augmented matrix as a system of equations. 1 0 −3 −4 0 1 2 0 If a matrix has m rows and n columns, we say the order of the matrix is m × n. What are the orders of the above matrices? In order to solve a system, we need to “reduce” the matrix to a form where we can identify the solution. . It is equivalent to the original system, This form is called just simplified. Reducing Matrices using Your Calculator: Step 1: Enter the matrix: Press 2nd x−1 . Scroll to the right to EDIT. Scroll down to your desired matrix (say, 1:[A]). Press EN T ER . Enter the dimensions of your matix, pressing EN T ER after each dimension. Enter each entry reading left to right and top to bottom, pressing EN T ER after each entry. Press 2nd M ODE to return to the home screen. Step 2: Press 2nd x−1 . Scroll to MATH. Step 3: Select rref(. Press EN T ER . Step 4: Press 2nd x−1 . Select your desired matrix ([A]). Step 5: Press ). Step 6: Press EN T ER . Once you have the reduced form of the matrix, you can determine the solution by rewriting each row back as an equation. Example 7. Solve the following systems of equations. 2x − y − z = 0 a) 3x + 2y + z = 7 x + 2y + 2z = 5 4 3x − 9y + 6z = −12 b) x − 3y + 2z = −4 2x − 6y + 4z = 8 3x − 4y = 10 c) −5x + 8y = −17 −3x + 12y = −12 In general, if you have more equations than variables, any solution is possible (unique, no solutions, or infinitely many solutions.) −x + 2y + 3z = 14 2x − y + 2z = 2 d) −x + 5y + 11z = 44 When a system has infinitely many solutions, we (write the solution in the solution ): A particular solution (or specific solution) to the system is a soultion found by choosing a value for the parameter. What are some particular solutions for this system? 5 Example 8. Solve the following system of equations. 7x + 2y − 2z − 4w + 3v = 8 −3x − 3y + 2w + v = −1 4x − y − 8z + 20v = 1 In general, if you have more variables than equations, then you will either have no solution or infinitely many solutions. 6 Example 9. The following reduced matrices represent systems of equations with variables x, y, z, and, if necessary, u. Determine the solutions to these systems. 1 0 0 0 3 0 1 0 0 −4 a) 0 0 1 2 3 0 0 0 0 1 1 0 b) 0 0 0 1 0 0 0 6 0 −9 1 0 0 0 0 1 0 0 2 5 0 0 1 0 c) 0 0 1 0 2 −1 6 0 1 −3 4 1 1 0 0 0 0 0 1 3 d) e) 0 −5 0 0 1 2 0 0 7 Systems of equations are used to model problems. For the following examples, set up a system of equations to solve the problem and solve. ALWAYS DEFINE YOUR VARIABLES when setting up a system of equations. Example 10. You have a total of $4000 on deposit with two savings institutions. Institution A pays simple interest at the rate of 3% per year, whereas Institution B pays simple interest at the rate of 4% per year. If you earn a total of $125 in interest during a single year, how much do you have on deposit in each institution? Example 11. 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. They brought in a total of $3750 that day, and the money made from basketballs alone was 4 times the money made from volleyballs alone. How many footballs, basketballs, and volleyballs were sold? 8 Example 12. (From Tan #36 2.3) A dietician wishes to plan a meal around three foods. The meal is to include 880 units of vitamin A, 3380 units of vitamin C, and 1020 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table: Vitamin A Vitamin C Calcium Food I 400 110 90 Food II 1200 570 30 Food III 800 340 60 Determine the amount of each food the dietician should include in the meal in order to meet the vitamin and calcium requirements. Just set up the problem. Example 13. (Tomastik/Epstein #40, 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 the table 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. Let x, y, and z represent the number of loungers, chairs, and footstools made respectively. Lounger Chair Footstool Wood Fabric 40 40 30 20 20 10 9 Stuffing 20 20 10 If a system of equations arising from a word problem has a parametric solution, sometimes there really aren’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 and that x must be a whole number [an integer]. Example 14. (Problem #49, Section 4.4) A person has 36 coins made up of nickels, dimes, and quarters. If the total value of the coins is $4. How many of each type of coin does this person have? Let x, y, and z be the number of nickels, dimes, and quarters the person has respectively. 10 Example 15. An instructor wants to write a quiz with 9 questions where each question is worth 3, 4, or 5 points based on difficulty. He wants the number of 3-point questions to be 1 more than the number of 5-point questions, and he wants the quiz to be worth a total of 35 points. How many 3, 4, and 5 point questions could there be? Let x, y, and z be the number of 3-point, 4-point, and 5-point questions respectively. 11
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