Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.Sparrowtown Photography Studio is taking graduation portraits for students at local schools. Eighth graders from Sparrowtown Elementary School ordered 53 basic portrait packages and 77 deluxe portrait packages, for a total of $10,350. The seniors at Salem High ordered 71 basic portrait packages and 96 deluxe portrait packages, for a total of $13,150. How much does each type of package cost?A basic package costs $_____, and a deluxe package costs $_____.
Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.Sparrowtown Photography Studio is taking graduation portraits for students at local schools. Eighth graders from Sparrowtown Elementary School ordered 53 basic portrait packages and 77 deluxe portrait packages, for a total of $10,350. The seniors at Salem High ordered 71 basic portrait packages and 96 deluxe portrait packages, for a total of $13,150. How much does each type of package cost?A basic package costs $_____, and a deluxe package costs $_____.
Set Up Equations: Let's denote the cost of a basic package as b and the cost of a deluxe package as d. We need to set up two equations based on the information given.Eighth graders ordered 53 basic and 77 deluxe packages for $10,350, which gives us the equation:53b+77d=10,350Seniors ordered 71 basic and 96 deluxe packages for $13,150, which gives us the equation:71b+96d=13,150
Solve Using Elimination: We now have a system of two equations with two variables:53b+77d=10,35071b+96d=13,150We can solve this system using either substitution or elimination. Let's use the elimination method to solve for one of the variables.
Eliminate Variable 'b': To eliminate one of the variables, we can multiply the first equation by 71 and the second equation by 53, so that the coefficients of 'b' in both equations are the same.(53b+77d)×71=10,350×71(71b+96d)×53=13,150×53This gives us:3763b+5467d=735,3503763b+5088d=697,950
Solve for 'd': Now we subtract the second equation from the first to eliminate 'b':(3763b+5467d)−(3763b+5088d)=735,350−697,950This simplifies to:379d=37,400
Substitute to Solve for 'b': We can now solve for 'd' by dividing both sides of the equation by 379: d=37937,400d=98.68Since the cost of the packages should be a whole number, we can round 'd' to the nearest whole number, which is $99.
Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute d back into one of the original equations to solve for b. Let's use the first equation:53b+77d=10,35053b+77(99)=10,35053b+7623=10,350
Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute d back into one of the original equations to solve for b. Let's use the first equation:53b+77d=10,35053b+77(99)=10,35053b+7623=10,350 Subtract 7623 from both sides to solve for b:53b=10,350−762353b=2727b=532727b=51.45Since the cost of the packages should be a whole number, we can round b to the nearest whole number, which is b1.
Calculate Cost of Packages: Now that we have the cost of the deluxe package, we can substitute d back into one of the original equations to solve for b. Let's use the first equation:53b+77d=10,35053b+77(99)=10,35053b+7623=10,350 Subtract 7623 from both sides to solve for b:53b=10,350−762353b=2727b=532727b=51.45Since the cost of the packages should be a whole number, we can round b to the nearest whole number, which is b1.We have found the cost of both types of packages:A basic package costs b1, and a deluxe package costs b3.
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