Saturday, May 21, 2016

Daily Regents: Modeling Systems of Equations

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 36

36. An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday.

Write an equation to represent to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat's numbers possible? Use your equation to justify your answer. Later, Pat found a record sowing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?

This is a lot of work for a system of equations.

We'll use C for the number of cats and D for the number of dogs. That makes 2.35C the amount spent on cats and 5.50D is the amount spent on dogs.

The equation for Wednesday is

2.35C + 5.50D = 89.50.

For the second part, you want to see if Pat's numbers could be correct. The way to do that would be to put them into the equation and see if it works. Substitute 8 for C and 14 for D, and check if it's a true statement.

2.35(8) + 5.50(14) = 89.50 ?
18.80 + 77.00 = 89.50 ?
95.80 =/= 89.50 X

This is NOT a possible combination.

The final part states that there were 22 cats and dogs altogether. So the total number of cats (C) plus the number of dogs D is 22. You can write that as an equation

C + D = 22.

We now have a system of equations that we can solve simultaneously.

Multiply this second equation by -2.35 to match the coefficient of C in the first equation. You can solve using Elimination.

-2.35C + -2.35D = -51.70
2.35C + 5.50D = 89.50
combine the two equations: 3.15D = 37.80
divide by 3.15: D = 12

There are 12 Dogs, so there are 22 - 12 = 10 = C, 10 Cats.

Any questions?

If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.

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