SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 765 Determine a, b, and c so that each equation is satisfied. That is, solve the system of three equations containing three variables: − + =− + + = + + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ a b c a b c a b c 4 6 9 3 0 (1) (2) (3) Solving this system of equations, we obtain = − = a b 2, 5, and = c 3. So the quadratic function whose graph contains the points ( ) ( ) − − 1, 4 , 1,6 ,and ( ) 3, 0 is = − + + y x x 2 5 3 2 y ax bx c a b c , 2, 5, 3 2 = + + =− = = Figure 7 shows the graph of the function, along with the three points. Figure 7 y x x 2 5 3 2 =− + + x y –4 –2 2 4 (3, 0) (1, 6) (–1, –4) 2 4 6 –5 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 11.1 Assess Your Understanding 1. Solve the equation: + = − x x 3 4 8 . (pp. A45–A47) 2. (a) Graph the line: + = x y 3 4 12. (b) What is the slope of a line parallel to this line? (pp. 32–43) 3. True or False If a system of equations has no solution, it is said to be dependent. 4. If a system of equations has one solution, the system is and the equations are . 5. If the only solution to a system of two linear equations containing two variables is = = − x y 3, 2, then the graphs of the lines in the system intersect at the point . 6. If the lines that make up a system of two linear equations are coincident, then the system is and the equations are . Concepts and Vocabulary 7. Multiple Choice If a system of two linear equations in two variables is inconsistent, then the graphs of the lines in the system are . (a) intersecting (b) parallel (c) coincident (d) perpendicular 8. Multiple Choice If a system of dependent equations containing three variables has the general solution x y z x z y z z , , 4, 2 5, is any real number ( ) { } =− + =− + then is one of the infinite number of solutions of the system. (a) ( ) − 1, 1, 3 (b) ( ) 0, 4, 5 (c) ( ) − 4, 3, 0 (d) ( ) −1, 5, 7 Skill Building In Problems 9–18, verify that the values of the variables listed are solutions of the system of equations. 9. − = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 5 5 2 8 ( ) = = − − x y 2, 1; 2, 1 10. + = − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 3 2 2 7 30 ( ) = − = − x y 2, 4; 2, 4 11. − = − =− ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ x y x y 3 4 4 1 2 3 1 2 ( ) = = x y 2, 1 2 ; 2, 1 2 12. + = − =− ⎧ ⎨ ⎪⎪ ⎪⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ x y x y 2 1 2 0 3 4 19 2 ( ) = − = − x y 1 2 , 2; 1 2 , 2 13. x y x y 3 1 2 3 − = + = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ( ) = = x y 4, 1; 4, 1 14. x y x y 3 3 1 − = − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ( ) = − = − − − x y 2, 5; 2, 5 15. + + = − − = − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z y z 3 3 2 4 0 2 3 8 = = − = x y z 1, 1, 2; ( ) − 1, 1, 2 16. − = + − = − − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x z x y z x y z 4 7 8 5 0 5 6 = = − = x y z 2, 3, 1; ( ) − 2, 3, 1 17. + + = − + = − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 3 3 2 4 3 10 5 2 3 8 ( ) = = − = − x y z 2, 2, 2; 2, 2,2 18. − = − =− − − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x z y z x y z 4 5 6 5 17 6 5 24 ( ) = = − = − x y z 4, 3, 2; 4, 3,2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Now Work PROBLEM 73
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