SECTION 11.2 Systems of Linear Equations: Matrices 769 89. Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables. 90. Do you prefer the method of substitution or the method of elimination for solving a system of two linear equations containing two variables? Give your reasons. 88. Make up a system of three linear equations containing three variables that has: (a) No solution (b) Exactly one solution (c) Infinitely many solutions Give the three systems to a friend to solve and critique. Explaining Concepts ‘Are You Prepared?’ Answers 1. 1{ } 2. (a) x y 4 (0, 3) (4, 0) –2 –2 2 2 (b) 3 4 − Problems 91–100 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 91. Graph ( ) = − + − f x 3 2. x 1 92. Factor each of the following: (a) x x x x x 4 2 3 2 5 2 5 3 2 3 3 3 2 3 2 4 ( ) ( ) ( ) ( ) − ⋅ ⋅ + + + ⋅ ⋅ − (b) ( ) ( ) ( ) ( ) − ⋅ ⋅ + − + − − − − x x x x 1 2 3 5 3 3 1 2 3 3 5 1 2 1 2 3 2 1 2 97. If = π ⋅ z e6 i 7 4 and = π ⋅ w e2 , i 5 6 find zw and z w . Write the answers in polar form and in exponential form. 98. Find the principal needed now to get $5000 after 18 months at 4% interest compounded monthly. 99. Find the average rate of change of ( ) = − f x x cos 1 from = − x 1 2 to = x 1 2 . 100. Find the area of the triangle with vertices at ( ) 0, 5 , ( ) 3, 9 , and ( ) 12, 0 . 93. Find the exact value of π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − sin sin 10 9 . 1 94. Write − +i 3 in polar form and in exponential form. 95. If … { } = A 2, 4, 6, , 30 and … { } = B 3, 6, 9, , 30 , find ∩A B. 96. Find an equation of an ellipse if the center is at the origin, the length of the major axis is 20 along the x -axis, and the length of the minor axis is 12. OBJECTIVES 1 Write the Augmented Matrix of a System of Linear Equations (p. 770) 2 Write the System of Equations from the Augmented Matrix (p. 771) 3 Perform Row Operations on a Matrix (p. 771) 4 Solve a System of Linear Equations Using Matrices (p. 773) 11.2 Systems of Linear Equations: Matrices The systematic approach of the method of elimination for solving a system of linear equations provides another method of solution that involves a simplified notation. Consider the following system of linear equations: + = − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 4 14 3 2 0 If we choose not to write the variables, we can represent this system as − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 3 4 2 14 0 where it is understood that the first column represents the coefficients of the variable x, the second column the coefficients of y, and the third column the constants
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