Chapter 6.7, Problem 60SR

To determine

To find: the number of gold medals won by U.S

Answer to Problem 60SR

The solution to the system of equation is $\left(9,9\right)$

Total number of gold medal won by U.S is $9$

Total number of silver medal won by U.S is $9$

Explanation of Solution

Given:

Total number of gold and silver medal won by U.S in $2006$ was $18$

Total points scored for gold and silver medal by U.S in $2006$ was $45$

Calculation:

Let $x$ be the total number gold medals won by U.S

Let $y$ be the total number gold medals won by U.S

Therefore total number of medals won by U.S is $x+y$

This gives $x+y=18$

Points scored for gold medals are $3x$ and points scored for silver medals are $2y$

Thus total points are $3x+2y$

Therefore $3x+2y=45$

Therefore system of equation is $\begin{array}{l}x+y=18\\ 3x+2y=45\end{array}$

Place the coefficients of the equations and the constant term into a matrix

  $\begin{array}{l}x+y=18\\ 3x+2y=45\end{array}$$\to$

  $\left[\begin{array}{l}1118\\ 3245\end{array}\right]$

Therefore the augmented matrix is

  $\left[\begin{array}{l}1118\\ 3245\end{array}\right]$

To make the first element in the second row a $0$ , multiply first row by $-3$ and add the result to the second row

  $\left[\begin{array}{l}1118\\ 3245\end{array}\right]$$\left[\begin{array}{l}1118\\ 0-1-9\end{array}\right]$

To make the second element in the second row a $1$ , multiply second row by $-1$

  $\left[\begin{array}{l}1118\\ 0-1-9\end{array}\right]$$\left[\begin{array}{l}1118\\ 019\end{array}\right]$

To make the second element in the first row a $0$ , multiply second row by $-1$ and add the result to the first row

  $\left[\begin{array}{l}1118\\ 019\end{array}\right]$$\left[\begin{array}{l}109\\ 019\end{array}\right]$

Hence,

The solution to the system of equation is $\left(9,9\right)$

Total number of gold medal won by U.S is $9$

Total number of silver medal won by U.S is $9$

Conclusion:

Hence, the solution to the system of equation is $\left(9,9\right)$