Question

I need help with this question parts a - d

Answer 1

A.

You have to assign the variables to the universities, I will do it like this:

x=Austin

y=Miami

z=California

B.

Now interpret the statement and write the equations:

Austin has 3 times the students that Miami has.:

x=3y

Berkley has 3000 students more than Miami twice:

z=3000+2y

The sum of all is 96000

x+y+z=96000

Now let all the variables on the left side of the equations and the constants on the right side. So the equations system looks like this:

x-3y = 0

2y-z = -3000

x+y+z = 96000

C.

Set the matrix:

$\begin{pmatrix}1&-3&0\\ 0&2&-1\\ 1&1&1\end{pmatrix}\begin{pmatrix}0\\ -3000\\ 96000\end{pmatrix}$

now let's subtract row 1 from row 3

$F_3-1*F_1->F_3$$\begin{pmatrix}1&-3&0\\ 0&2&-1\\ 0&4&1\end{pmatrix}\begin{pmatrix}0\\ -3000\\ 96000\end{pmatrix}$

now let's subtract 2 times row 2 from row 3

$F_3-2*F_2->F_3$$\begin{pmatrix}1 & -3 & 0 \\ 0 & 2 & -1 \\ 0 & 0 & 3\end{pmatrix}\begin{pmatrix}0 \\ -3000 \\ 102000\end{pmatrix}$

Now we can formulate whit the third line:

$3z=102000$$z=\frac{102000}{3}=34000$

Now replace Z in F2:

$2y-z=-3000$$2y-34000=-3000$$2y=31000$$y=\frac{31000}{2}=15500$

Now replace y in F1:

$x-3y=0$$x-3(15500)=0$$x=3(15500)=46500$

So we have:

x=46500 students

y=15500 students

z=34000 students

D.

We can interpret that:

The University of Texas has 46500 students, The University of Miami has 15500 and the University of California has 34000 students.