Question

The University of Texas at Austin has three times as many students enrolledas the University of Miami. The University of California, Berkley has 3,000 more than twice the number of students as the University of Miami. If the three schools have a total enrollment of 96,000 students, what is the enrollment at each school?

Define the variables in this problem (there are 3 variables).2.Write the system of linear equations to represent the scenario.3.Write an augmented matrix for the system of linear equations.4.Solve the system of equations using your augmented matrix. Write down all of the row operations you perform.5.Using at least one complete sentence, answer the question: What is the enrollment at each school?​

Answer 1

Answer :-

University of Texas has 46,500 students, University of California has 34,000 students and University of Miami has 15,500 students.

Explanation :-

Represent Texas with T, Miami with M and California with C. With the given information, the following equations were obtained.

T = 3 × M

T – 3M = 0 ………………………….(i)

C = 3,000 + 2 × M

C – 2M = 3,000…………………..(ii)

T + C + M = 96,000…………….(iii)

By rearranging, we have:

1T + 0C – 3M = 0 ………………………….(i)

0T + 1C – 2M = 3,000…………………..(ii)

1T + 1C + 1M = 96,000………….…….(iii)

The Augmented Matrix is:

$\left[\begin{array}{ccc}1&0&-3\\0&1&-2\\1&1&1\end{array}\right]\left[\begin{array}{ccc}T\\C\\M\end{array}\right] = \left[\begin{array}{ccc}0\\3,000\\96,000\end{array}\right]$

Making $\left[\begin{array}{ccc}T\\C\\M\end{array}\right]$ the subject of the formula we have:

$\left[\begin{array}{ccc}T\\C\\M\end{array}\right] = \frac{\left[\begin{array}{ccc}0\\3,000\\96,000\end{array}\right]}{\left[\begin{array}{ccc}1&0&-3\\0&1&-2\\1&1&1\end{array}\right]}$

Let’s say matrix A = $\left[\begin{array}{ccc}1&0&-3\\0&1&-2\\1&1&1\end{array}\right]$, the inverse of matrix A = $\left[\begin{array}{ccc}\frac{1}{2} &-\frac{1}{2} &\frac{1}{2} \\-\frac{1}{3} &\frac{2}{3} &\frac{1}{3} \\-\frac{1}{6} &-\frac{1}{6} &\frac{1}{6} \end{array}\right]$

is determined by the formula $\sf{\frac{Adjoint(A)}{determinant(A)}.}$This can also be computed

directly on a scientific calculator.

$\left[\begin{array}{ccc}T\\C\\M\end{array}\right] = \left[\begin{array}{ccc}\frac{1}{2} &-\frac{1}{2} &\frac{1}{2} \\-\frac{1}{3} &\frac{2}{3} &\frac{1}{3} \\-\frac{1}{6} &-\frac{1}{6} &\frac{1}{6} \end{array}\right] \left[\begin{array}{ccc}0\\3,000\\96,000\end{array}\right]= \left[\begin{array}{ccc}46,500\\34,000\\15,500\end{array}\right]$

T = 46,500

C = 34,000

M = 15,500