202 Holt McDougal Algebra 1

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1 a. Let x represent the number of minutes and y represent the cost. Long Distance Inc.: y = 0.0x Far Away Calls: y = 0.0x + 1. Graph y = 0.0x and y = 0.0x Cost of Long Distance Calls Cost ($) Far Away Calls (7, 1.66) Long Distance Inc x c 6-4c c -4-9 c c x x ( x ) ( ) x Minutes The solution appears to be at (7, 1.66). So, the cost for both companies will be the same for 7 min and that cost will be $1.66. b. It is better to use Long Distance Inc. if the call is under 7 minutes because it costs less. If the call is over 7 minutes, it is better to use Far Away Calls because it costs less. c. The line will intersect the y-axis at 1.. Far Away Calls is cheaper because it costs less per minute. SPIRAL REVIEW. 1 = 7 x 7 (1) = 7 ( 4 = x 7 x ). -6y = y -6 = y = x = 1 - ( - x ) = -(1) x = = y 1 1 ( ) = 1 ( y 4 4 = y 1) 7. numbers less than. numbers greater than or equal to 7 9. numbers greater than (x + 1) > 4(x) + 4(1) > x + 4 > x > 4 x > 4 x > 6- SOLVING SYSTEMS BY SUBSTITUTION, PAGES CHECK IT OUT! 1a. y = x + x + = x + x + = - - x = - y = x + y = - + y = 1 (-, 1) c. x + y = -4 -x -x y = -x - 4 x + y = -7 x + (-x - 4) = -7 -x - 4 = x = - -1(-x) = -1(-) x = x + y = -7 + y = y = -10 (, -10) b. x + y = 16 (y - 4) + y = 16 10y - 4 = y = 0 10y 10 = 0 10 y = x = y - 4 x = () - 4 x = 4-4 x = 0 (0, ) 0 Holt McDougal Algebra 1

2 . -x + y = +x +x y = x + x + y = 9 x + (x + ) = 9 x + (x) + () = 9 x + 4x + 16 = 9 7x + 16 = x = -7 7x 7 = -7 7 x = -1 -x + y = -(-1) + y = + y = - - y = 6 (-1, 6) a. Let m represent the number of months and let t represent the total cost. Option 1: t = m Option : t = m t = m m = m - 70m - 70m 160 = m = 10m = 10m = m t = m = (10) = = 60 In 10 months, the total cost for each option will be the same, $60. b. The first option; the first option is cheaper for the first 9 months; the second option is cheaper after 10 months. THINK AND DISCUSS 1. at the point (, 10). The solution for each is (, ). EXERCISES GUIDED PRACTICE 1. y = x - 10 x + = x x -x = x = x 1 = x 9 = x y = x - 10 y = (9) - 10 y = 4-10 y = (9, ). 4x + y = 0 4x + (x + ) = 0 x + = x = 1 x = 1 x = y = x + y = + y = (, ). x + y = -x -x y = -x + 4x + y = 0 4x + (-x + ) = 0 x + = x = 1 x + y = (1) + y = 4 + y = -4-4 y = - (1, -) 0 Holt McDougal Algebra 1

3 4. x - y = 10 + y +y x = y + 10 x - y = 4 (y + 10) - y = 4 (y) + (10) - y = 4 x - y = 10 x - (1) = 10 x - = x = 1 (1, 1) y + - y = 4 - y = y - 4x = + 4x +4x y = 4x + -y = -1 -(-y) = -1(-1) y = 1 x - y = 1 x - (4x + ) = 1 x - (4x) - () = 1 x - 1x - 9 = 1-10x - 9 = x = 0-10x -10 = 0-10 x = - y - 4x = y - 4(-) = y + 1 = y = -9 (-, -9) 6. -x - y = 0 -(y - ) - y = 0 -(y) - (-) - y = 0 -y + - y = 0 - y = y = - -y - = - - y = 4 x = y - x = 4 - x = -4 (-4, 4) 7a. Let m represent the number of months and let t represent the total cost. Green Lawn: t = m Grass Team: t = + 7m t = m + 7m = m - 9m - 9m + m = m = 4 m = 4 m = t = m = () = = 16 In months, the total cost for each company will be the same, $16. b. Green Lawn; for 6 months, Green Lawn s service costs only (6) = = $, while Grass Team s costs + 7(6) = + = $47. PRACTICE AND PROBLEM SOLVING. y = x + 9. y = x + 10 x + 4 = x + -x - 6 = x x -x +x +x x + 4 = -6 = 4x x = = 4x y = x = 4x 4 y = = x y = (-1, ) y = x + 10 y = (-4) + 10 y = y = (-4, ) 10. x + y = - y -y x = -y + x + y = 1 (-y + ) + y = 1 y + = y = 4 x + y = x + (4) = x + = - - x = 0 (0, 4) 04 Holt McDougal Algebra 1

4 11. x + y = - y -y x = -y + x = -y + x = -y + 1-4x + 4y = 1-4(-y + 1) + 4y = 1-4(-y) - 4(1) + 4y = 1 4y y = 1 y - 4 = y = 16 y = 16 y = x + y = x + () = x + 4 = x = - x = - x = -1 (-1, ) 1. -y = -x + 4 -(0.x + ) = -x + 4 -(0.x) - () = -x x - = -x + 4 +x +x 1.x - = x = 6 1.x 1. = 6 1. x = 4 y = 0.x + y = 0.(4) + y = + y = 4 (4, 4) 1. -x + y = 4 +x +x y = x + 4 x - y = -7 x - (x + 4) = -7 x - (x) - (4) = -7 x - x - = -7 x - = x = 1 -x + y = 4 -(1) + y = y = y = (1, ) 14. x + y = - -x -x y = -x - -x - y = 6 -x - (-x - ) = 6 -x - (-x) - (-) = 6 -x + x + = 6 x + = x = - x + y = - (-) + y = y = y = - (-, -) 1. x + y = -1 - y -y x = -y - 1 4x - 4y = 0 4(-y - 1) - 4y = 0 4(-y) + 4(-1) - 4y = 0 -y - 4-4y = 0-1y - 4 = y = 4-1y -1 = 4-1 y = - x + y = -1 x + (-) = -1 x - 4 = x = (, -) 16. 4x = y x + 1 = y 6x - y = - 6x - (4x + 1) = - 6x - (4x) - (1) = - 6x - x - = - -x - = x = -1 -x - = -1 - x = 4x = y ( ) = y - 1 = y = y, ) ( 0 Holt McDougal Algebra 1

5 17a. Let m represent the number of months and let t represent the total cost. Option 1: t = 10 + m Option : t = 60m t = 10 + m 60m = 10 + m -m - m m = 10 m = 10 m = 6 t = 60m = 60(6) = 60 In 6 months, the total cost for each option will be the same, $60. b. the second option; for months, it will cost only 60() = $00, while the other option costs 10 + () = = $. 1. x = x + y = + y = - - y = (, ) 19. x = y + 6 x = (-x + 4) + 6 x = (-x) + (4) + 6 x = -6x x = -6x x +6x 7x = 14 7x 7 = 14 7 x = y = -x + 4 y = -() + 4 y = y = - (, -) 0. x - y = 11 -x -x -y = -x (-y) = -1(-x + 11) y = -1(-x) - 1(11) y = x - 11 y - 7x = 1 (x - 11) - 7x = 1 (x) + (-11) - 7x = 1 1x - - 7x = 1 x - = x = 6 x = 6 x = 7 x - y = 11 (7) - y = y = y = -10-1(-y) = -1(-10) y = 10 (7, 10) 1. x - y = + y +y x = y + x + y = 6 (y + ) + y = 6 (y) + () + y = 6 y y = 6 x - y = x - 6 = x = (, 6) 6 y + 1 = y = 6 ( 6 y ) 6 = () y = 6 06 Holt McDougal Algebra 1

6 . x + y = (7 - y) + y = (7) + (-y) + y = 14-4y + y = 14 - y = y = -9 -y - = -9 - y = x = 7 - y x = 7 - () x = 7-6 x = 1 (1, )..x + = y.x + = 1.x x -1.x x + = x = -9 y = 1.x - 4 y = 1.(-9) - 4 y = y = -14. (-9, -14.) 4. Let x represent the first number and let y represent the second number. Sum is 0: x + y = 0 First is 4 less than twice second: x = y - 4 x + y = 0 x = y - 4 x + y = 0 (y - 4) + y = 0 y - 4 = y = 9 y = 9 y = 1 x = y - 4 x = (1) - 4 x = 6-4 x = 19 The two numbers are 19 and coins: n + d = 0 Value $1.40: 0.0n d = 1.40 n + d = 0 -n -n d = -n n d = n (-n + 0) = n (-n) (0) = n n +.00 = n +.00 = n = n -0.0 = n = 1 n + d = d = d = There are 1 nickels and dimes in the jar. 6. Let p represent the price of the popcorn and d represent the price of the drinks. Customer #9: p + d = 1 Customer #99: p + 4d = p + d = 1 p + 4d = p + 4d = - 4d -4d p = -4d + p = -4d + p = -d + 11 p + d = 1 (-d + 11) + d = 1 (-d) + (11) + d = 1-6d + + d = 1-4d = d = -1-4d -4 = -1-4 d = p + d = 1 p + () = 1 p + 6 = p = 1 p = 1 p = The price of a large bag of popcorn is $ and the price of a small drink is $. 07 Holt McDougal Algebra 1

7 7. Let x represent the amount in the % account and let y represent the amount in the 6% account. Total of $1000: x + y = 1000 $ in interest: 0.0x y = x + y = x y = x + y = y -y x = -y x y = 0.0(-y ) y = 0.0(-y) + 0.0(1000) y = -0.0y y = y = y = 0.01y 0.01 = 0.01 y = 00 x + y = 1000 x + 00 = x = 00 Helen invested $00 in the % account and $00 in the 6% account.. x + y = 90 x + (4x - 10) = 90 x - 10 = x = 100 x = 100 m x = 0 y = 4x - 10 y = 4(0) - 10 y = 0-10 m y = x + y = 90 x + (x - 1) = 90 x + (x) + (-1) = 90 x + x - 0 = 90 x - 0 = x = 10 x = 10 m x = 40 y = (x - 1) y = (40-1) y = () m y = 0 9. x + y = 90 y + y = 90 y = 90 y = 90 m y = 0 x = y x = (0) m x = 60 m y = 0 b. (p - w) = 40 (p) + (-w) = 40 p - w = 40 p - w = 40 p + w = 40 (p + w) = 40 (p) + (w) = 40 p + w = 40 c. p - w = 40 + w +w p = w + 40 p = w + 40 p = w + 0 p + w = 40 (w + 0) + w = 40 (w) + (0) + w = 40 w w = 40 4w = w = 0 4w 4 = 0 4 w = 0 p + w = 40 p + (0) = 40 p + 40 = p = 00 p = 00 p = 100 The speed of the plane is 100 mi/h and the speed of the wind is 0 mi/h.. Possible answer: Solve one of the equations for either x or y. Then substitute the expression equal to x or y into the other equation. This creates a onevariable equation that can be solved. When you get the value of one variable, substitute it into one of the original equations and solve to find the value of the other variable.. The solution of a system solved by graphing is the same as the solution of a system solved by substitution. 4a. Let x represent the cost of a book and let y represent the cost of a backpack. Total $6: x + y = 6 Book costs $ less than backpack: x = y - x + y = 6 x = y - 1a. Rate Time = Distance With Headwind p - w = 40 With Tailwind p + w = 40 0 Holt McDougal Algebra 1

8 b. x + y = 6 (y - ) + y = 6 (y) + (-) + y = 6 y y = 6 y - 16 = y = 4 y = 4 y = 14 x = y - x = 14 - x = 6 book: $6; backpack: $14 c. x + y = 6 -x -x y = -x + 6 Cost of Backpack ($) Juanita s Purchase (6, 14) x = y x + = y Cost of Book ($) The solution appears to be at (6, 14). Possible answer: Substitution works well since x is already isolated. Graphing requires solving both equations for y.. Possible estimate: (1.7, -.) x + y = x -x y = -x x - y = 6 x - (-x - 0.6) = 6 x - (-x) - (-0.6) = 6 x + x = 6 x = x =.4 x =.4 x = 1. x + y = y = y = -.4 (1., -.4) 7. F; If d is the number of dimes and n is the number of nickels, then d + n represents the number of coins, so Roger has 1 coins. Since d is greater than n, there are more dimes than nickels, so F is correct. CHALLENGE AND EXTEND. Let n represent the number of new cars and let u represent the number of used cars. Total of 7 cars: n + u = 7 Ratio of new to used is :4: 4 = n u 4n = u n + u = 7 4n = u n + u = 7 -n -n u = -n + 7 4n = u 4n = (-n + 7) 4n = (-n) + (7) 4n = -n n +n 9n = 190 9n 9 = n = 10 n + u = u = u = 16 The car dealership has 10 new cars and 16 used cars. 9. t = 4 s + t = 10 s + (4) = 10 s + 1 = s = - r - s - t = 1 r - (-) - 4 = 1 r = 1 r + = r = 10 r = 10 r = r = ; s = -; t = 4 TEST PREP 6. D; Since the total number of cousins is 4, m + f = 4 or f = 4 - m. Since the number of males was 6 less than twice the number of females, m = f - 6, so D is correct. 09 Holt McDougal Algebra 1

9 40. y + z = -y -y z = -y + y - 4z = -14 y - 4(-y + ) = -14 y - 4(-y) - 4() = -14 y + 4y - 0 = -14 6y - 0 = y = 6 6y 6 6 = 6 y = 1 y + z = 1 + z = -1-1 z = 4 x + y + z = 7 x = 7 x + = x = x = ; y = 1; z = b + c = - +b +b c = b - b + c = 1 b + (b - ) = 1 b + (b) + (-) = 1 b + b - 10 = 1 b - 10 = b = b = b = -b + c = c = c = 0 a + b + c = 19 a + () + 0 = 19 a + 10 = a = 9 a = 9; b = ; c = 0 SPIRAL REVIEW 4. Possible answer: The hedge grows for a while, then someone trims it, and then it starts to grow again. 4. Possible answer: The weather is bad for the first days, so few people are at the beach. Then the temperature warms up for the next days and the number of visitors increases. For the last days, the weather is warm, but cloudy, so the number of visitors drops. 44. Possible answer: A motorcycle begins accelerating in a residential area and maintains a slow speed until it gets to the highway. Then it accelerates to a faster speed, which it maintains for a while until it brakes and slows to a complete stop. 4. 6x - y = 1 6x - (0) = 1 6x - 0 = 1 6x = 1 6x 6 = 1 6 x = The x-intercept is y + x = 1 -(0) + x = x = 1 x = 1 The x-intercept is y - 40 = -x y = -x x +x 4y + x = 40 x + 4y = 40 x + 4(0) = 40 x + 0 = 40 x = 40 x = 40 x = The x-intercept is. 4. x - y = -6 () - (0) (, 0) is not a solution of the system. 49. y - x = 6 (4) - (-1) 6 (4) (-1, 4) is a solution of the system. 6x - y = 1 6(0) - y = y = 1 -y - = 1 - y = -6 The y-intercept is -6. -y + x = 1 -y + 0 = 1 -y = 1 -y - = 1 - y = - The y-intercept is -. x + 4y = 40 (0) + 4y = y = 40 4y = 40 4y 4 = 40 4 y = 10 The y-intercept is 10. x + y = () + (0) x + 4y = 1 (-1) + 4(4) 1 (-1) y + x = 7 x = 1 () 1 (6) + () (, 6) is not a solution of the system. 10 Holt McDougal Algebra 1