Question

I need a little help with the second part of b and part c! + can y'all check my explanations please

a. (my answer for context) The system for this word problem is x+y=14; 11.99x+24.99y=245.86 (x is shirts and y is pants)

b. 1. Explain, without solving, which method is most efficient in solving this system: substitution, elimination, graphing, or making a table?
(my answer is substitution, because the solutions to x and y can easily be substituted into the first equation).

b. 2. Explain why the other methods are not as sufficient.
(I need a little help but I'm thinking it's because for this system solving method you basically only have to solve the second equation and then just substitute)

c. How could you simplify the numbers used in this system to simplify the system? Does this new system change your answer to part b? Explain. (What does this mean? How can I answer it?)

Answer 1

b. Your answer is appropriate.

c. The decimals can be eliminated by multiplying by 100. (You can also add 0.14 to the problem, adding .01 to each individual price and 0.14 to the total. This will make the second equation be 12x +25y = 246, which makes the numbers slightly easier to work with.) Unfortunately, there don't seem to be any common factors that would reduce the size of the numbers.

Explanation:

a. We assume you're finding quantities of shirts and pants for a purchase of 14 total items costing $245.86. If so, your equations are appropriate.

b. The solution for one variable or the other can be found in 6 arithmetic operations using your suggested method. Any other method takes as many or more. (Multiplying the first equation by -11.99 and adding to the second gets you the equivalent of substituting for x in as many operations.)

Graphing can be quick and easy with the right tool. Since you know the solutions are integers, just about any graphing tool will get you close enough.

Another method (using only 5 arithmetic operations) is to calculate the average price of an item as 245.86/14, then figuring the ratio of the difference from the lower price to the total difference in price. Multiplying this ratio by the number of items gives the number of the higher-priced item (pants). (245.86/14 -11.99)/(24.99 -11.99)·14 = 6 This method treats the problem as a "mixture" problem, finding the numbers of each consitutuent that make the average come out what it is.

c. See above for how to simplify the numbers. Done properly, the answer is not changed. (Some people don't work well with decimals or fractions, so like to see those eliminated from the problem. Some folks work better with small integers than with large ones, so like to see the numbers be made as small as possible.)