Chapter 6.3, Problem 31PPS

a.

To determine

To write the system of equations that represents the number of U.S. teams and non-U.S. teams.

Answer to Problem 31PPS

  $\begin{array}{l}x+y=66\\ x-y=30\end{array}$

Explanation of Solution

Given:

Number of professional mountain bike racing teams: 66

Number of non-U.S. teams is 30 more than the number of U.S. teams.

Calculation:

Let x represent the number of non-U.S. teams and y represent the number of U.S. teams.

The total number of professional mountain bike racing teams is 66.

So, it can be represented as:

  $x+y=66\to \text{(1)}$

Also, the number of non-U.S. teams is 30 more than the number of U.S. teams. Then,

  $\begin{array}{l}x=y+30\\ \Rightarrow x-y=30\to \text{(2)}\end{array}$

Conclusion:

Therefore, the equations that represent the U.S. and Non-U.S. teams are

  $\begin{array}{l}x+y=66\\ x-y=30\end{array}$

b.

To determine

To find the solution to the system of equations found in (a).

Answer to Problem 31PPS

  $x=48$

  $y=18$

Explanation of Solution

Given:

From subpart (a),

The system of equations is:

  $\begin{array}{l}x+y=66\\ x-y=30\end{array}$

Calculation:

Let the equations be labelled as follows:

  $\begin{array}{l}x+y=66\to \text{(1)}\\ x-y=30\to (2)\end{array}$

Add the above equations.

  $\begin{array}{l}x+y=66\\ \frac{x-y=30}{2x=96}\end{array}$

  $\begin{array}{c}2x=96\\ x=\frac{96}{2}\\ x=48\end{array}$

Substitute $x=48$ in any of the equations.

Say, Equation (2),

  $\begin{array}{c}48-y=30\text{ [Subtracting 48 on both sides]}\\ -y=30-48\\ -y=-18\\ y=18\end{array}$

Conclusion:

Therefore, the solutions of the system of equations is $x=48$ and $y=18.$

c.

To determine

To interpret the solution as the context of the question

Answer to Problem 31PPS

Non-U.S. teams: 48

U.S. teams: 18

Explanation of Solution

Given:

From subpart (b),

  $x=48$

  $y=18$

Calculation:

Let x represent the number of non-U.S. teams and y represent the number of U.S. teams.

Then the number of non-U.S. teams is 48

The number of U.S. teams is 18.

Conclusion:

Therefore, the number of non-U.S. teams and U.S. teams are 48 and 18 respectively.

d.

To determine

Graph the system of equations and check the solution.

Explanation of Solution

Given:

From subpart (a),

The system of equations is:

  $\begin{array}{l}x+y=66\\ x-y=30\end{array}$

Graph:

Using a graphing utility the system of equations is graphed as follows:

Interpretation:

From the above graph, the point of intersection of x and y co-ordinates is the same as the solutions found algebraically.

Conclusion:

Therefore, the solutions of the equations are verified graphically.