Find the nth term of -5,-2,3,10,19
1 answer:
You might be interested in
9514 1404 393
Answer:
- 10 study all three
- 8 study only math and chem
Step-by-step explanation:
1. Adding the numbers of those who study a given subject counts ...
- once -- those who study only one subject
- twice -- those who study exactly two subjects
- three times -- those who study all three subject.
That total is 40 +48 +44 = 132. When we subtract from this the total number of students, we get 132 -80 = 52, the number who study exactly 2 subjects plus twice the number who study all three.
From this, we can subtract the number who study exactly 2 subjects, leaving twice the number who study all three. 52 -32 = 20. So, 10 students study all three subjects.
__
2. Adding the numbers of those who study physics and another subject, we have 20+24 = 44. Subtracting twice the number who study all three gives the total of those how study only two subjects, one of which is physics. That total is 44 -20 = 24. So, of those 32 who study exactly two subjects, the remaining 32 -24 = 8 must be students who study only math and chemistry.
_____
It may be helpful to see the matrix of equations the problem statement describes. If we use the variables m, p, c, mp, mc, pc, mpc in that order to represent numbers of 1-subject, 2-subject, and 3-subject students, the augmented matrix looks like this.
The solution is m=20, p=6, c=12, mp=10, mc=8, pc=14, mpc=10. The bold values are the ones this question is asking about.
To find an inverse matrix you first have to find the determinant and then put the absolute value of it under a 1. That's what is sitting outside in front of all those matrices. What you have to do now is multiply that determinant by what I teach to my students as the "mixed up matrix". They always remember from that. Change the position of the terms on the major axis (here they are a and d), and then switch the signs on the other 2 terms. Our inverse matrix then is ![A^{-1} =\frac{1}{IAI} \left[\begin{array}{ccc}d&-b\\-c&a\\\end{array}\right]](https://tex.z-dn.net/?f=%20%20A%5E%7B-1%7D%20%3D%5Cfrac%7B1%7D%7BIAI%7D%20%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dd%26-b%5C%5C-c%26a%5C%5C%5Cend%7Barray%7D%5Cright%5D%20)