.6 Solving Linear Systems Using Technology Essential Question How can you represent algebraic expressions using a coefficient matrix? A matrix is a rectangular arrangement of numbers. The dimensions of a matrix with m rows and n columns are m n. So, the dimensions of matrix A are 2. A = 4 6 5 2 rows columns Writing Coefficient Matrices Work with a partner. Match each set of algebraic expressions with its coefficient matrix. Explain your reasoning. Sample Algebraic 2x + y + 6z Coefficient 2 expressions: x + z matrix: a. 4x + y b. 4x + z 5x + y 5x + y c. 4x z d. 4x + 2y +z 5x y y z 4z 6 A. 4 5 C. 4 5 B. 4 D. 4 5 2 4 Writing Coefficient Matrices ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, you need to analyze mathematical relationships to connect mathematical ideas. Work with a partner. Write and enter the coefficient matrix for each set of expressions into a graphing calculator. a. 2y 5z b. 5y z c. x y x + y 2x + 4z x + z x + 2y + 9z Communicate Your Answer. How can you represent algebraic expressions using a coefficient matrix? 4. Write the algebraic expressions that are represented by the coefficient matrix. MATRIXA 4 4-4 5-8 4-9 Section.6 Solving Linear Systems Using Technology 4
.6 Lesson What You Will Learn Core Vocabulary matrix, p. 42 dimensions of a matrix, p. 42 elements of a matrix, p. 42 augmented matrix, p. 42 Previous system of three linear equations Write augmented matrices for systems of linear equations. Use technology to solve systems of linear equations in three variables. Writing Augmented Matrices for Systems A matrix is a rectangular arrangement of numbers. The dimensions of a matrix with m rows and n columns are m n (read m by n ). So, the dimensions of matrix A are 2. The numbers in the matrix are its elements. A = 4 6 columns 5 2 rows The element in the first row and third column is 5. A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the system. For example, the system below can be represented by the given augmented matrix. System: 5x + y 2z = Augmented x + 2y + z = matrix: x 4y + z = 5 coefficients 2 4 2 constants Before you write an augmented matrix, make sure each equation in the system is written in standard form. Include zeros for the coefficients of any missing variables. This determines the order of the constants and coefficients in the augmented matrix. Writing an Augmented Matrix Write an augmented matrix for the system. Then state the dimensions. SOLUTION y 2x = x + y = 2 Begin by rewriting each equation in the system in standard form. 2x + y = x + y = 2 Next, use the coefficients and constants as elements of the augmented matrix. 2 2 The augmented matrix has two rows and three columns, so the dimensions are 2....... 42 Chapter Linear Functions, Linear Systems, and Matrices
Writing an Augmented Matrix COMMON ERROR Because the second equation does not have a z-term, the coefficient of z is. Write an augmented matrix for the system. Then state the dimensions. x 4y = 7 2z 9x + y = 2x + 4y z = 2 SOLUTION Begin by rewriting each equation in the system in standard form. x 4y + 2z = 7 9x + y + z = 2x + 4y z = 2 Next, use the coefficients and constants as elements of the augmented matrix. 4 2 7 9 2 4 2 The augmented matrix has three rows and four columns, so the dimensions are 4. Monitoring Progress Write an augmented matrix for the system. Then state the dimensions.. x 8y = 4 2. x + y = z. 9x 8y + z = 5x + 2y = 9 7x + 9y z = 2 x y + 2z = 6 6x + 4y + 8z = x + 4y = 6 Solving Systems of Equations Using Technology Many technology tools have matrix features that you can use to solve a system of linear equations. The augmented matrix in Example 2, rewritten in reduced row-echelon form, is shown below. Observe that the solution to this system is (x, y, z) = (.4, 5.2, 6). You can verify this solution in the original system..4 5.2 6 Core Concept Solving a Linear System Using Technology Step Write an augmented matrix for the linear system. Step 2 Enter the augmented matrix into your graphing calculator. Step Use the reduced row-echelon form feature to rewrite the system. Step 4 Interpret the result from Step to solve the linear system. Section.6 Solving Linear Systems Using Technology 4
Solving a System Using Technology Use a graphing calculator to solve the system. x + y + z = 2 2x + y + z = x y 2z = 6 REMEMBER An m n matrix has m rows and n columns. SOLUTION Step Write an augmented matrix for the linear system. 2 2 2 6 Step 2 Enter the dimensions and elements of the augmented matrix into your graphing calculator. NAMES :A 2:B :C 4:D 5:E 6:F 7:G MATH EDIT MATRIXA 4 2 2 - -2-6 Step Use the reduced row-echelon form feature to rewrite the system. NAMES MATH 6:randM( 7:augment 8:Matr list( 9:List matr( :cumsum A:ref( B:rref( EDIT rref(a - 2 Step 4 Converting the matrix back to a system of linear equations, you have: x = y = z = 2 The solution is x =, y =, and z = 2, or the ordered triple (,, 2). Check this solution in each of the original equations. Check x + y + z = 2 2x + y + z = x y 2z = 6 + + 2 =? 2 2() + () + 2 =? 2(2) =? 6 2 = 2 = 6 = 6 Monitoring Progress Use a graphing calculator to solve the system. 4. x + 2y z = 2 5. x + y z = 6. x + 2y 5z = x y + z = 2x 6y + z = 6x z = 8 x + 4y 4z = 4 x + 5y 2z = 4 y + z = 2 44 Chapter Linear Functions, Linear Systems, and Matrices
.6 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check. COMPLETING THE SENTENCE A matrix derived from a system of linear equations is called the matrix of the system. 2. WRITING Describe how to find the solution of a system of linear equations in three variables using technology. Monitoring Progress and Modeling with Mathematics In Exercises, write an augmented matrix for the system. Then state the dimensions. (See Examples and 2.). x 4y = 7 4. 5y 4x = 7 9x + y = x 7y = 5 5. x + 8y 7z = 2 6. 4x 5y + 2z = 5x + 9y + 5z = 5 6x + 4y + 9z = 8 6z y 8x = x 2y z = 7 7. x y + z = 4 8. 5x + 2z = 9 6x 5z = x + 5y 8z = 5 x + 7y + 8z = 5 4x + 2y + 9z = 9. x + 2y = z + 7. x + y = 5z + 9 5x + 4z = 8y 2x 4y + 5z = 2x + 9y z = 6 4y + z = 6x ERROR ANALYSIS In Exercises and 2, describe and correct the error in writing an augmented matrix for the system below.. 2. x 9y + 5z = 8 2x + y = 5 6x 9z + 4y = 4 The augmented matrix is: 9 5 8 2 5 6 9 4 4 The augmented matrix is: 9 5 8 2 5 6 4 9 4 In Exercises 22, use a graphing calculator to solve the system. (See Example.). 4x + y + 6z = 7 4. x + 4y z = 7 x + y + 2z = 7 2x y + 2z = 5 x y + z = 9 x + y z= 22 5. x + y + z = 9 6. x 2y + z = 9 x + y + z = 2x + 5y + z = 5x 2z = x 6y + 9z = 2 7. x + z = 6 8. 4x + y + 6z = 2 2x + y + z = 2x + 2y + 4z = x y + 2z = x y + z = 5 9. x + y + 4z = 7 2. x + y + 2z = 2x y z = 24 x y + z = 4x + 2y + 2z = 8 x + y + 6z = 4 2. x + y + 2z = 22. 2x + 4y + z = x + 2y + z = 5 x y z = 2 x + 4y + z = 5 5x y z = 8 2. MODELING WITH MATHEMATICS A company sells three types of gift baskets. The basic basket has two movie passes and one package of microwave popcorn, and costs $5.5. The medium basket has two movie passes, two packages of microwave popcorn, and one DVD, and costs $7. The super basket has four movie passes, three packages of microwave popcorn, and two DVDs, and costs $72.5. a. Write an augmented matrix to represent b. Use a graphing calculator to find the cost of each basket item. Section.6 Solving Linear Systems Using Technology 45
24. MODELING WITH MATHEMATICS You go shopping at a local department store with your friend and cousin. You buy one pair of jeans, four pairs of shorts, and two shirts for $84. Your friend buys two pairs of jeans, one pair of shorts, and three shirts for $76. Your cousin buys one pair of jeans, two pairs of shorts, and one shirt for $52. a. Write an augmented matrix to represent b. Use a graphing calculator to find the cost of each piece of clothing. 25. MODELING WITH MATHEMATICS You have 85 coins in nickels, dimes, and quarters with a combined value of $.25. There are twice as many quarters as dimes. a. Write an augmented matrix to represent b. Use a graphing calculator to find the number of each type of coin. 26. HOW DO YOU SEE IT? Write a system of equations for the augmented matrix below. 2 4 8 9 2 4 6 27. MAKING AN ARGUMENT Your friend states that the number of rows in an augmented matrix of the system will always be the same as the number of variables in the system. Is your friend correct? Explain your reasoning. 28. USING TOOLS Use a graphing calculator to solve the system of four linear equations in four variables. 2w + 5x 4y + 6z = 2x + 2y 7z = 52 4w + 8x 7y + 4z = 25 w + 6x 5y + z = 6 29. REASONING Is it possible to write more than one augmented matrix for a system of linear equations? Explain your reasoning.. MATHEMATICAL CONNECTIONS The sum of the measures of the angles in ABC is 8. The sum of the measures of angle B and angle C is twice the measure of angle A. The measure of angle B is 2 less than the measure of angle C. a. Write an augmented matrix to represent b. Use a graphing calculator to find the measures of the three angles.. ABSTRACT REASONING Let a, b, and c be real numbers. Classify the linear system represented by each matrix as consistent or inconsistent. Explain your reasoning. a. b. c. a b c a b a b 2. THOUGHT PROVOKING Write an augmented matrix, not in reduced row-echelon form, for a system that has exactly one solution, (x, y, z) = (2,, 5). Justify your answer. Maintaining Mathematical Proficiency Solve the inequality. Graph the solution.. 2 x > 5 4. 5z + 8 7 5. 2w + 7 w + 5 6. 2 r + < 2 r + 7 Reviewing what you learned in previous grades and lessons Graph the inequality in a coordinate plane. 7. y < x 8. y 4 > 2x + 6 9. 2 x + y 5 4. x y 9 46 Chapter Linear Functions, Linear Systems, and Matrices