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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineSparrowtown Photography Studio is taking graduation portraits for students at local schools. Eighth graders from Sparrowtown Elementary School ordered 5353 basic portrait packages and 7777 deluxe portrait packages, for a total of $10,350\$10,350. The seniors at Salem High ordered 7171 basic portrait packages and 9696 deluxe portrait packages, for a total of $13,150\$13,150. How much does each type of package cost?\newlineA basic package costs $\$_____, and a deluxe package costs $\$_____.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineSparrowtown Photography Studio is taking graduation portraits for students at local schools. Eighth graders from Sparrowtown Elementary School ordered 5353 basic portrait packages and 7777 deluxe portrait packages, for a total of $10,350\$10,350. The seniors at Salem High ordered 7171 basic portrait packages and 9696 deluxe portrait packages, for a total of $13,150\$13,150. How much does each type of package cost?\newlineA basic package costs $\$_____, and a deluxe package costs $\$_____.
  1. Set Up Equations: Let's denote the cost of a basic package as bb and the cost of a deluxe package as dd. We need to set up two equations based on the information given.\newlineEighth graders ordered 5353 basic and 7777 deluxe packages for $10,350\$10,350, which gives us the equation:\newline53b+77d=10,35053b + 77d = 10,350\newlineSeniors ordered 7171 basic and 9696 deluxe packages for $13,150\$13,150, which gives us the equation:\newline71b+96d=13,15071b + 96d = 13,150
  2. Solve Using Elimination: We now have a system of two equations with two variables:\newline53b+77d=10,35053b + 77d = 10,350\newline71b+96d=13,15071b + 96d = 13,150\newlineWe can solve this system using either substitution or elimination. Let's use the elimination method to solve for one of the variables.
  3. Eliminate Variable 'b': To eliminate one of the variables, we can multiply the first equation by 7171 and the second equation by 5353, so that the coefficients of 'b' in both equations are the same.\newline(53b+77d)×71=10,350×71(53b + 77d) \times 71 = 10,350 \times 71\newline(71b+96d)×53=13,150×53(71b + 96d) \times 53 = 13,150 \times 53\newlineThis gives us:\newline3763b+5467d=735,3503763b + 5467d = 735,350\newline3763b+5088d=697,9503763b + 5088d = 697,950
  4. Solve for 'd': Now we subtract the second equation from the first to eliminate 'b':\newline(3763b+5467d)(3763b+5088d)=735,350697,950(3763b + 5467d) - (3763b + 5088d) = 735,350 - 697,950\newlineThis simplifies to:\newline379d=37,400379d = 37,400
  5. Substitute to Solve for 'b': We can now solve for 'd' by dividing both sides of the equation by 379379: \newlined=37,400379d = \frac{37,400}{379}\newlined=98.68d = 98.68\newlineSince the cost of the packages should be a whole number, we can round 'dd' to the nearest whole number, which is $99\$99.
  6. Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute dd back into one of the original equations to solve for bb. Let's use the first equation:\newline53b+77d=10,35053b + 77d = 10,350\newline53b+77(99)=10,35053b + 77(99) = 10,350\newline53b+7623=10,35053b + 7623 = 10,350
  7. Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute dd back into one of the original equations to solve for bb. Let's use the first equation:\newline53b+77d=10,35053b + 77d = 10,350\newline53b+77(99)=10,35053b + 77(99) = 10,350\newline53b+7623=10,35053b + 7623 = 10,350 Subtract 76237623 from both sides to solve for bb:\newline53b=10,350762353b = 10,350 - 7623\newline53b=272753b = 2727\newlineb=272753b = \frac{2727}{53}\newlineb=51.45b = 51.45\newlineSince the cost of the packages should be a whole number, we can round bb to the nearest whole number, which is bb11.
  8. Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute dd back into one of the original equations to solve for bb. Let's use the first equation:\newline53b+77d=10,35053b + 77d = 10,350\newline53b+77(99)=10,35053b + 77(99) = 10,350\newline53b+7623=10,35053b + 7623 = 10,350 Subtract 76237623 from both sides to solve for bb:\newline53b=10,350762353b = 10,350 - 7623\newline53b=272753b = 2727\newlineb=272753b = \frac{2727}{53}\newlineb=51.45b = 51.45\newlineSince the cost of the packages should be a whole number, we can round bb to the nearest whole number, which is bb11.We have found the cost of both types of packages:\newlineA basic package costs bb11, and a deluxe package costs bb33.

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