Chapter 6.2, Problem 1MP

(a)

To determine

To define two variables to represent the amounts of time Ashley can spend on the two exercise machines.

Explanation of Solution

Let’s define x as the number of minutes Ashley spends on the stair machine and y as the number of minutes Ashley spends on the rowing machine.

(b)

To determine

To determine a system of two equations that describe the relationship between the amounts of time Ashley spends on two machines.

Answer to Problem 1MP

  $x+y=40$ and $x=2y$

Explanation of Solution

Given information:

1. From part (a) x represents the number of minutes Ashley spends on the stair machine and y represents the number of minutes Ashley spends on the rowing machine.
2. Ashley will exercise for 40 minutes, diving her time between the stair machine and the rowing machine.
3. Ashley will spend twice as much time on the stair machines as on the rowing machine.

Calculation:

From information number (ii), we get that the total time Ashley spends on the both stair machine and rowing machine is 40.

That is, sum of x and y is 40

That is, $x+y=40$

From information number (iii), we get that the time Ashley spends on stair machine is double that of rowing machine.

That is, value of x will be double that of y .

That is, $x=2y$

∴The system of two equations that describe the relationship between the amounts of time Ashley spends on two machine are:

  $x+y=40$ and $x=2y$

(c)

To determine

Solve the system of two equations in (b).

Answer to Problem 1MP

  $x=26.667$ and $y=13.33$

Explanation of Solution

Given information:

From part (b) the system of two equations that describe the relationship between the amounts of time Ashley spends on two machine are:

  $x+y=40$ and $x=2y$

Formula Used:

Substitution Method:

The substitution method is the algebraic method to solve system of linear equation. In this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the equation gets transformed into one linear equation with only one variable, which can be easily solved.

Calculation:

  $x+y=40$ ......... (1)

  $x=2y$ .......... (2)

Substitute $x=2y$ in $x+y=40$

∴ (1)$\Rightarrow$

  $2y+y=40$

  $\Rightarrow$$y\left(2+1\right)=40$

  $3y=40$

  $\Rightarrow$Dividing both side of above equation by 3

  $\frac{3y}{3}=\frac{40}{3}$

  $\Rightarrow$$y=13.33$

Substitute $y=13.33$ in $x=2y$

∴ (2)$\Rightarrow$

  $\begin{array}{c}x=2\times 13.33\\ =26.667\end{array}$

Solution of system of equations $x+y=40$ and $x=2y$ is:

  $x=26.667$ and $y=13.33$

(d)

To determine

Interpret the solution of the system found in part (c)

Explanation of Solution

Given information:

Solution of the system of equation $x+y=40$ and $x=2y$ is:

  $x=26.667$ and $y=13.33$

From part (a),

x represents the number of minutes Ashley spends on the stair machine and y represents the number of minutes Ashley spends on the rowing machine

  $x=26.667$ means that the time Ashley spends on the stair machine is 26.667 minutes.

  $y=13.33$ means that the time Ashley spends on the rowing machine is 13.33 minutes