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Chapter 5 Review 3 1. Is (−2 , ) a solution to 4 ⎧⎪ x − 12y = −11 ⎨ −2x + 8y = 10 ⎩⎪ ! 3⎞ 3 ⎛ ⎜⎝ −2, ⎟⎠ means x = −2 and y = 4 4 3 into both 4 Equation A and Equation B to see if they work in both equations plug x = −2 and y = Equation A ⎧ x − 12y = −11 ⎨ Equation B ⎩−2x + 8y = 10 Equation A (x) − 12(y) = −11 Equation B − 2(x) + 8(y) = 10 ⎛ 3⎞ −2 − 12 ⎜ ⎟ = −11 ⎝ 4⎠ ⎛ 3⎞ − 2 (−2) + 8 ⎜ ⎟ = 10 ⎝ 4⎠ −2 − 9 = −11 −11 = −11 4 + 6 = 10 10 = 10 3⎞ ⎛ ⎜⎝ −2, ⎟⎠ works in both equations so 4 YES it is a solution. Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel Solve each system of equations by graphing. List your answers as an ordered pair. Line A ⎧⎪ y = x + 2 3. ⎨ Line B ⎪ y = −2x + 8 ⎩ ! The lines intersect at the point ( 3 , 4 ) Line B (x,y) Line A ⎧ −2 x+2 Line A ⎪ 2x + 3y = 6 ⇒ y = 5. ! 3 ⎨ Line B ⎪ y=4 ⎩ The lines intersect at the point ( –3 , 4 ) ! (x,y) !! ! ! ! Line B Line A Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel Solve each system of equations. List your answers as an ordered par. 7. Equation A ⎧⎪ −4x + 2y = 20 ! ⎨ Equation B ⎪ 3x + y = −5 ⎩ Multiply Equation B by − 2 so that Equation A has 2y and Equation B has − 2y ⎧−4x + 2y = 20 ⎨ −2 ⎩ 3x + y = −5 − 4x + 2y = 20 − 6x − 2y = 10 − 10x = 30 x= −3 Plug x = −3 into either equation A and solve for y Equation A −4(x) + 2y = 20 −4(−3) + 2y = 20 12 + 2y = 20 2y = 8 y=4 Answer: (− 3, 4 ) Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 8. Equation A ⎧ x − 2y = −5 ⎨ Equation B ⎩2x − 4y = 6 Multiply Equation A by − 2 so that Equation A has –2x and Equation B has − 2x −2 ⎧ x − 2y = −5 ⎨ ⎩2x − 4y = 6 − 2x + 4y = 10 2x − 4y = 6 0 = 16 STOP: Both the x and y terms canceled out and the remaining equation 0 = −2 is false The lines are parallel,they have no common points. Answer: No Solution Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 11. Equation A ⎧⎪ 2x + 4y = −6 ! ⎨ Equation B ⎪ x = 2y − 5 ⎩ Equation B has been solved for x. Substitute the expression x is equal to (2y − 5) into Equation A in place of y. Equation A with 2y − 5 subtituted for x 2(2y − 5) + 4y = −6 Solve for y 4y − 10 + 4y = −6 8y − 10 = −6 8y = 4 y= 1 2 ! 1 into Equation B to find x 2 x = 2(y) − 5 plug y = ⎛ 1⎞ x = 2⎜ ⎟ − 5 ⎝ 2⎠ x = 1− 5 x = −4 1⎞ ⎛ Answer: ⎜ −4, ⎟ ⎝ 2⎠ Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 12. Equation A ⎧⎪ 8x − 4y = 20 ! ⎨ Equation B ⎪ y = 2x − 5 ⎩ Equation B has been solved for y. Substitute the expression x is equal to (2x − 5) into Equation A in place of y. Equation A with 2x − 5 subtituted for y 8x − 4(2x − 5) = 20 Solve for y 8x − 8x + 20 = 20 20 = 20 Stop: The x term canceled out and the remaining eqution 20 = 20 is true Both equations describe the same line any point on 8x − 4y = 20 would also be on y = 2x − 5 Answer: All Points on 8x − 4y = 20 or Answer: All Points on y = 2x − 5 either one of the above is correct Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 14. Equation A ⎧⎪ 2x − 4y = 10 ⎨ Equation B ⎪ −x + 2y = −5 ⎩ Multiply Equation B by 2 to eliminate the x terms ⎧⎪ 2x − 4y = 10 ⎨ 2 ⎪ −x + 2y = −5 ⎩ 2x − 4y = 10 −2x + 4y = −10 0=0 STOP: Both the x and y terms canceled out and the remaining equation 0 = 0 is true Both equations describe the same line any point on 2x − 4y = 10 would also be on − x + 2y = −5 Answer: All Points on 2x − 4y = 10 or Answer: All Points on − x + 2y = −5 either one of the above is correct Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 15. Equation A ⎧⎪ 4x + 2y = 5 ⎨ Equation B ⎪ 2x + y = −1 ⎩ Multiply Equation B by − 2 to eliminate the x terms ⎧ 4x + 2y = 5 ⎨ −2 ⎩2x + y = −1 4x + 2y = 5 −4x − 2y = −2 0=3 ! STOP: Both the x and y terms canceled out and the remaining equation 0 = 3 is false The lines are parallel , they have no common points. Answer: No Solution Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel Equation A ⎧⎪ 3x − 5y = 11 ⎨ Equation B ⎪ 2x − 6y = 2 ⎩ 17. You must mutiply both rows by different numbers to eliminate a variable Multiply Equation A by − 2 Multiply Equation B by 3 to eliminate the x terms −2 ⎧ 3x − 5y = 11 ⎨ 3 ⎩2x − 6y = 2 −6x + 10y = −22 6x − 18y = 6 − 8y = −16 y=2 Now add the two equations Solve for y Plug y = 2 into either equation A or B and solve for y Equation A 3(x) − 5(y) = 11 3(x) − 5(2) = 11 3x − 10 = 11 3x = 21 x=7 Answer: ( 7,2 ) Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 18. ⎧ x y + = −1 Equation A ⎪ 2 6 ⎪⎪ ⎨ Equation B ⎪ −x y −1 ⎪ + = ⎪⎩ 2 3 2 Multiply Equation A by 6 Multiply Equation B by 6 to eliminate the fractions 6 ⎧ x + y = −1 ⎪⎪ 2 6 1 ⎨ −x y −1 6 ⎪⎪ 2 + 3 = 2 ⎩ to get 2 new equations Equation C ⎧ 3x + y = −6 ⎨ Equation D ⎩−3x + 2y = −3 which you now can solve Add Equation C and Equation D to eliminate the x terms and solve for y Equation C ⎧ 3x + y = −6 ⎨ Equation D ⎩−3x + 2y = −3 3y = −9 y = −3 Plug y = −3 into either equation C or D and solve for x Equation C 3(x) + (y) = −6 3x − 3 = −6 3x = −3 x = −1 Answer: ( −1,−3) Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel 21. Tom runs and walks for a total of 24 miles. The number of miles he runs is three times the number of miles he walks. How far does he run and walk. Show all work Sentence Sentence Sentence Tom runs and walks for a total of 24 miles If we ADD the number of miles run(R) and walks (W) the sum is 24 R + W = 24 The number of miles he runs is three times the number of miles he walks The number of miles he runs (R) is = R= three times the number of miles he walks 3W R = 3W ⎧ R + W = 24 ⎨ ⎩ R = 3W substitutue the 3W for R in the top equation 3W + (W ) = 24 3W + W = 24 substitutue 3D for W solve for W 3W + W = 24 4W = 24 W =6 plug W = 4 into the second equation and solve for D R = 3W and W = 6 R = 3(6) R = 18 ANSWER : He runs 18 miles and walks 6 miles Check 18 + 6 = 24 18 = 3(6) Math 100 ! Chapter 5 Review WKD ! © 2016 Eitel