Comments
Description
Transcript
Solving Systems by Elimination
4.3. SOLVING SYSTEMS BY ELIMINATION 243 47. The second equation, y = −3 − 2x, is already solved for y. Substitute −3 − 2x for y in the first equation and solve for x. −9x + 4y = 73 −9x + 4(−3 − 2x) = 73 First Equation. Substitute −3 − 2x for y. −9x − 12 − 8x = 73 −12 − 17x = 73 Distribute the 4. Combine like terms. −17x = 85 x = −5 Add 12 to both sides. Divide both sides by −17. Finally, to find the y-value, substitute −5 for x in the equation y = −3 − 2x. y = −3 − 2x y = −3 − 2(−5) y = −3 + 10 Substitute −5 for x. Multiply. y=7 Simplify. Hence, (x, y) = (−5, 7) is the solution of the system. 4.3 Solving Systems by Elimination 1. Start with the given system. x + 9x − 4y 7y = = 0 −43 We’ll concentrate on eliminating x. Multiply the first equation by −9, then add the results. −9x − 36y = 0 9x − 7y = −43 − 43y = −43 Divide both sides by −43. −43 −43 y=1 y= Divide both sides by −43. Simplify. Take the answer y = 1 and substitute 1 for y in the first equation. x + 4y = 0 x + 4(1) = 0 x+4=0 x = −4 First equation. Substitute 1 for y. Multiply. Subtract 4 from both sides. Hence, the solution is (x, y) = (−4, 1). Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 244 Check: We must show that the solution (x, y) = (−4, 1) satisfies both equations. x + 4y = 0 9x − 7y = −43 (−4) + 4(1) = 0 −4 + 4 = 0 9(−4) − 7(1) = −43 −36 − 7 = −43 0=0 −43 = −43 Because each of the last two statements are true, this guarantees that (x, y) = (−4, 1) is a solution of the system. 3. Start with the given system. 6x + y 4x + 2y = = 8 0 We’ll concentrate on eliminating y. Multiply the first equation by −2, then add the results. −12x − 2y = −16 4x + 2y = 0 −8x = −16 Divide both sides by −8. −16 −8 x=2 x= Divide both sides by −8. Simplify. Take the answer x = 2 and substitute 2 for x in the first equation. 6x + y = 8 First equation. 6(2) + y = 8 12 + y = 8 Substitute 2 for x. Multiply. y = −4 Subtract 12 from both sides. Hence, the solution is (x, y) = (2, −4). Check: We must show that the solution (x, y) = (2, −4) satisfies both equations. 6x + y = 8 4x + 2y = 0 6(2) + (−4) = 8 12 − 4 = 8 4(2) + 2(−4) = 0 8−8=0 8=8 0=0 Second Edition: 2012-2013 4.3. SOLVING SYSTEMS BY ELIMINATION 245 Because each of the last two statements are true, this guarantees that (x, y) = (2, −4) is a solution of the system. 5. Start with the given system. −8x + 4x + y 3y = = −56 56 We’ll concentrate on eliminating y. Multiply the first equation by −3, then add the results. 24x − 3y = 168 4x + 3y = 56 28x = 224 Divide both sides by 28. 224 28 x=8 Divide both sides by 28. x= Simplify. Take the answer x = 8 and substitute 8 for x in the first equation. −8x + y = −56 −8(8) + y = −56 First equation. Substitute 8 for x. −64 + y = −56 y=8 Multiply. Add 64 to both sides. Hence, the solution is (x, y) = (8, 8). Check: We must show that the solution (x, y) = (8, 8) satisfies both equations. −8x + y = −56 −8(8) + (8) = −56 4x + 3y = 56 4(8) + 3(8) = 56 −64 + 8 = −56 32 + 24 = 56 −56 = −56 56 = 56 Because each of the last two statements are true, this guarantees that (x, y) = (8, 8) is a solution of the system. 7. Start with the given system. x + −5x − 8y 9y = = 41 −50 Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 246 We’ll concentrate on eliminating x. Multiply the first equation by 5, then add the results. 5x + 40y = 205 −5x − 9y = −50 31y = 155 Divide both sides by 31. 155 31 y=5 Divide both sides by 31. y= Simplify. Take the answer y = 5 and substitute 5 for y in the first equation. x + 8y = 41 First equation. x + 8(5) = 41 x + 40 = 41 Substitute 5 for y. Multiply. x=1 Subtract 40 from both sides. Hence, the solution is (x, y) = (1, 5). Check: We must show that the solution (x, y) = (1, 5) satisfies both equations. x + 8y = 41 −5x − 9y = −50 (1) + 8(5) = 41 1 + 40 = 41 −5(1) − 9(5) = −50 −5 − 45 = −50 41 = 41 −50 = −50 Because each of the last two statements are true, this guarantees that (x, y) = (1, 5) is a solution of the system. 9. Start with the given system. −12x + 9y −6x − 4y = = 0 −34 We’ll concentrate on eliminating x. Multiply the second equation by −2, then add the results. −12x + 9y = 0 12x + 8y = 68 17y Second Edition: 2012-2013 = 68 4.3. SOLVING SYSTEMS BY ELIMINATION 247 Divide both sides by 17. 17y = 68 68 y= 17 y=4 Divide both sides by 17. Simplify. Take the answer y = 4 and substitute 4 for y in the first equation. −12x + 9y = 0 −12x + 9(4) = 0 First equation. Substitute 4 for y. −12x + 36 = 0 −12x = −36 Multiply. Subtract 36 from both sides. Divide both sides by −12 x=3 Hence, the solution is (x, y) = (3, 4). 11. Start with the given system. 27x − −3x − 6y 5y = = −96 22 We’ll concentrate on eliminating x. Multiply the second equation by 9, then add the results. 27x − 6y = −96 −27x − 45y = 198 − 51y = 102 Divide both sides by −51. −51y = 102 102 y= −51 y = −2 Divide both sides by −51. Simplify. Take the answer y = −2 and substitute −2 for y in the first equation. 27x − 6y = −96 27x − 6(−2) = −96 27x + 12 = −96 27x = −108 x = −4 First equation. Substitute −2 for y. Multiply. Subtract 12 from both sides. Divide both sides by 27 Hence, the solution is (x, y) = (−4, −2). Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 248 13. Start with the given system. 2x − −3x + 6y 18y = = 28 −60 We’ll concentrate on eliminating y. Multiply the first equation by 3, then add the results. 6x − 18y = 84 −3x + 18y = −60 3x = 24 Divide both sides by 3. 3x = 24 24 x= 3 x=8 Divide both sides by 3. Simplify. Take the answer x = 8 and substitute 8 for x in the first equation. 2x − 6y = 28 2(8) − 6y = 28 First equation. Substitute 8 for x. 16 − 6y = 28 −6y = 12 Multiply. Subtract 16 from both sides. y = −2 Divide both sides by −6 Hence, the solution is (x, y) = (8, −2). 15. Start with the given system. −32x + 7y 8x − 4y = −238 = 64 We’ll concentrate on eliminating x. Multiply the second equation by 4, then add the results. −32x + 7y = −238 32x − 16y = 256 − 9y = 18 Divide both sides by −9. −9y = 18 18 y= −9 y = −2 Second Edition: 2012-2013 Divide both sides by −9. Simplify. 4.3. SOLVING SYSTEMS BY ELIMINATION 249 Take the answer y = −2 and substitute −2 for y in the first equation. −32x + 7y = −238 −32x + 7(−2) = −238 First equation. Substitute −2 for y. −32x − 14 = −238 −32x = −224 Multiply. Add 14 to both sides. Divide both sides by −32 x=7 Hence, the solution is (x, y) = (7, −2). 17. Start with the given system. 3x − −2x − 7y 2y = = −75 −10 We’ll first concentrate on eliminating the variable x. Multiply the first equation by 2, the second equation by 3, then add the results. 6x − −6x − 14y 6y = = −150 −30 − 20y = −180 Divide both sides by −20. −20y = −180 −180 y= −20 y=9 Divide both sides by −20. Simplify. Take the answer y = 9 and substitute 9 for y in the first equation (you could also make the substitution in the second equation). 3x − 7y = −75 3x − 7(9) = −75 3x − 63 = −75 3x = −12 −12 x= 3 x = −4 First equation. Substitute 9 for y. Multiply. Add 63 to both sides. Divide both sides by 3. Simplify. Hence, the solution is (x, y) = (−4, 9). Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 250 19. Start with the given system. 9x − 9y 2x − 6y = = −63 −34 We’ll first concentrate on eliminating the variable x. Multiply the first equation by −2, the second equation by 9, then add the results. −18x + 18x − 18y 54y = = 126 −306 − 36y = −180 Divide both sides by −36. −36y = −180 −180 y= −36 y=5 Divide both sides by −36. Simplify. Take the answer y = 5 and substitute 5 for y in the first equation (you could also make the substitution in the second equation). 9x − 9y = −63 First equation. 9x − 9(5) = −63 9x − 45 = −63 Substitute 5 for y. Multiply. 9x = −18 −18 x= 9 x = −2 Add 45 to both sides. Divide both sides by 9. Simplify. Hence, the solution is (x, y) = (−2, 5). 21. Start with the given system. −9x − 2y 5x − 3y = 28 = −32 We’ll first concentrate on eliminating the variable x. Multiply the first equation by −5, the second equation by −9, then add the results. 45x + −45x + Second Edition: 2012-2013 10y 27y = = −140 288 37y = 148 4.3. SOLVING SYSTEMS BY ELIMINATION 251 Divide both sides by 37. 37y = 148 148 y= 37 y=4 Divide both sides by 37. Simplify. Take the answer y = 4 and substitute 4 for y in the first equation (you could also make the substitution in the second equation). −9x − 2y = 28 −9x − 2(4) = 28 First equation. Substitute 4 for y. −9x − 8 = 28 −9x = 36 36 x= −9 x = −4 Multiply. Add 8 to both sides. Divide both sides by −9. Simplify. Hence, the solution is (x, y) = (−4, 4). 23. Start with the given system. −3x − 7x + 5y 7y = = −34 56 We’ll first concentrate on eliminating the variable x. Multiply the first equation by −7, the second equation by −3, then add the results. 21x + 35y −21x − 21y = = 238 −168 14y = 70 Divide both sides by 14. 14y = 70 70 y= 14 y=5 Divide both sides by 14. Simplify. Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 252 Take the answer y = 5 and substitute 5 for y in the first equation (you could also make the substitution in the second equation). −3x − 5y = −34 −3x − 5(5) = −34 First equation. Substitute 5 for y. −3x − 25 = −34 −3x = −9 −9 x= −3 x=3 Multiply. Add 25 to both sides. Divide both sides by −3. Simplify. Hence, the solution is (x, y) = (3, 5). 25. Start with the given system. 2x − 7x + 7y 6y −2 3 = = We’ll first concentrate on eliminating the variable x. Multiply the first equation by −7, the second equation by 2, then add the results. −14x + 14x + 49y 12y = = 14 6 61y = 20 Divide both sides by 61 to get y = 20/61. Next, we could substitute 20/61 for y in either equation and solve to find x. However, in this case it is probably easier to perform elimination again. Start with the given system again. 2x − 7x + 7y 6y −2 3 = = This time we concentrate on eliminating the variable y. Multiply the first equation by −6, the second equation by −7, then add the results. −12x + 42y −49x − 42y = = 12 −21 −61x = −9 Divide both sides by −61 to get x = 9/61. Hence, the solution is (x, y) = (9/61, 20/61). Check: First, store 9/61 in X with the following keystrokes. The result is shown in the first image below. 9 ÷ 6 Second Edition: 2012-2013 1 STO X, T, θ, n ENTER 4.3. SOLVING SYSTEMS BY ELIMINATION 253 Store 20/61 in Y with the following keystrokes. The result is shown in the first image below. 2 0 ÷ 6 1 STO ALPHA 1 ENTER Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is shown in the second image below. 2 × X, T, θ, n − × 7 ALPHA 1 ENTER Enter the right-hand side of the second equation with the following keystrokes. The result is shown in the second image below. 7 × X, T, θ, n + × 6 ALPHA 1 ENTER The result in the second image shows that 2x − 7y = −2 and 7x + 6y = 3 for x = 9/61 and y = 20/61. The solution checks. 27. Start with the given system. 2x + −5x + 3y 5y = = −2 2 We’ll first concentrate on eliminating the variable x. Multiply the first equation by 5, the second equation by 2, then add the results. 10x + −10x + 15y 10y = −10 = 4 25y = −6 Divide both sides by 25 to get y = −6/25. Next, we could substitute −6/25 for y in either equation and solve to find x. However, in this case it is probably easier to perform elimination again. Start with the given system again. 2x + −5x + 3y 5y = = −2 2 Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 254 This time we concentrate on eliminating the variable y. Multiply the first equation by −5, the second equation by 3, then add the results. −10x − −15x + 15y 15y −25x = = 10 6 = 16 Divide both sides by −25 to get x = −16/25. Hence, the solution is (x, y) = (−16/25, −6/25). Check: First, store −16/25 in X with the following keystrokes. The result is shown in the first image below. (-) 1 ÷ 6 2 5 STO X, T, θ, n ENTER Store −6/25 in Y with the following keystrokes. The result is shown in the first image below. (-) 6 ÷ 2 5 STO ALPHA 1 ENTER Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is shown in the second image below. × 2 X, T, θ, n + × 3 ALPHA 1 ENTER Enter the right-hand side of the second equation with the following keystrokes. The result is shown in the second image below. (-) 5 × X, T, θ, n + 5 × ALPHA 1 ENTER The result in the second image shows that 2x + 3y = −2 and −5x + 5y = 2 for x = −16/25 and y = −6/25. The solution checks. Second Edition: 2012-2013 4.3. SOLVING SYSTEMS BY ELIMINATION 255 29. Start with the given system. 9x + −7x − 4y 9y −4 3 = = We’ll first concentrate on eliminating the variable x. Multiply the first equation by 7, the second equation by 9, then add the results. 63x + −63x − 28y 81y = −28 = 27 − 53y = −1 Divide both sides by −53 to get y = 1/53. Next, we could substitute 1/53 for y in either equation and solve to find x. However, in this case it is probably easier to perform elimination again. Start with the given system again. 9x + −7x − 4y 9y −4 3 = = This time we concentrate on eliminating the variable y. Multiply the first equation by 9, the second equation by 4, then add the results. 81x + −28x − = −36 = 12 36y 36y = −24 53x Divide both sides by 53 to get x = −24/53. Hence, the solution is (x, y) = (−24/53, 1/53). Check: First, store −24/53 in X with the following keystrokes. The result is shown in the first image below. (-) 2 ÷ 4 5 3 STO X, T, θ, n ENTER Store 1/53 in Y with the following keystrokes. The result is shown in the first image below. 1 ÷ 5 3 STO ALPHA 1 ENTER Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is shown in the second image below. 9 × X, T, θ, n + 4 × ALPHA 1 ENTER Enter the right-hand side of the second equation with the following keystrokes. The result is shown in the second image below. Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 256 (-) × 7 − X, T, θ, n 9 × ALPHA 1 ENTER The result in the second image shows that 9x + 4y = −4 and −7x − 9y = 3 for x = −24/53 and y = 1/53. The solution checks. 31. Start with the given system. 2x + 2y 3x − 5y = = 4 3 We’ll first concentrate on eliminating the variable x. Multiply the first equation by −3, the second equation by 2, then add the results. −6x − 6x − 6y 10y = = −12 6 − 16y = −6 Divide both sides by −16 to get y = 3/8. Next, we could substitute 3/8 for y in either equation and solve to find x. However, in this case it is probably easier to perform elimination again. Start with the given system again. 2x + 2y 3x − 5y = = 4 3 This time we concentrate on eliminating the variable y. Multiply the first equation by 5, the second equation by 2, then add the results. 10x + 10y 6x − 10y = = 20 6 16x = 26 Divide both sides by 16 to get x = 13/8. Hence, the solution is (x, y) = (13/8, 3/8). Check: First, store 13/8 in X with the following keystrokes. The result is shown in the first image below. 1 3 ÷ Second Edition: 2012-2013 8 STO X, T, θ, n ENTER 4.3. SOLVING SYSTEMS BY ELIMINATION 257 Store 3/8 in Y with the following keystrokes. The result is shown in the first image below. 3 ÷ 8 STO 1 ALPHA ENTER Clear the calculator screen by pressing the CLEAR button, then enter the lefthand side of the first equation with the following keystrokes. The result is shown in the second image below. 2 × X, T, θ, n + × 2 ALPHA 1 ENTER Enter the right-hand side of the second equation with the following keystrokes. The result is shown in the second image below. 3 × X, T, θ, n − × 5 ALPHA 1 ENTER The result in the second image shows that 2x + 2y = 4 and 3x − 5y = 3 for x = 13/8 and y = 3/8. The solution checks. 33. Start with the given system. x + 7y −8x − 56y = = −32 256 We’ll concentrate on eliminating x. Multiply the first equation by 8, then add the results. 8x + 56y = −256 −8x − 56y = 256 0 = 0 Note that this last statement, 0 = 0, is true. Hence, the system has an infinite number of solutions. Second Edition: 2012-2013 CHAPTER 4. SYSTEMS 258 35. Start with the given system. 16x − −8x + 16y 8y = −256 = 128 We’ll concentrate on eliminating x. Multiply the second equation by 2, then add the results. 16x − 16y = −256 −16x + 16y = 256 0 = 0 Note that this last statement, 0 = 0, is true. Hence, the system has an infinite number of solutions. 37. Start with the given system. x − 4y 2x − 8y = = −37 54 We’ll concentrate on eliminating x. Multiply the first equation by −2, then add the results. −2x + 8y = 74 2x − 8y = 54 0 = 128 Note that this last statement, 0 = 128, is false. Hence, the system has no solution. 39. Start with the given system. x + 9y −4x − 5y = 73 = −44 We’ll concentrate on eliminating x. Multiply the first equation by 4, then add the results. 4x + 36y = 292 −4x − 5y = −44 31y = 248 Divide both sides by 31. 31y = 248 248 y= 31 y=8 Second Edition: 2012-2013 Divide both sides by 31. Simplify.