Chapter 6.4, Problem 38PFA

a.

To determine

To solve the system of equations and find the coordinates of city A

Answer to Problem 38PFA

  $(-1,1)$

Explanation of Solution

Given:

Path of Carolyn’s flight: $y=-3x-2$

Path of Jay’s flight: $y=2x+3$

Calculation:

Consider both the equations,

  $\begin{array}{l}y=2x+3\\ \Rightarrow -2x+y=\mathrm{3..................}(1)\end{array}$

  $\begin{array}{l}y=-3x-2\\ \Rightarrow 3x+y=-\mathrm{2..................}(2)\end{array}$

Subtract equation (1) and equation (2),

  $\begin{array}{l}\Rightarrow (-2x+y)-(3x+y)=3+2\\ \Rightarrow -5x=5\\ \Rightarrow x=-1\end{array}$

Substitute the value of x in the given equation,

  $\begin{array}{c}y=2(-1)+3\\ =-2+3\\ y=1\end{array}$

Conclusion:

The coordinates of city A are $(-1,1).$

b.

To determine

To find the equation of Jay and Carolyn’s path to city B

Answer to Problem 38PFA

  $x+y=2$

Explanation of Solution

Given:

From subpart (a), the point is (-1, 1).

Jay and Carolyn fly from city A to city B with a slope -1.

Formula used:

Point slope formula: $(y-y_1)=m(x-x_1)$

Calculation:

Let $(x_1,y_1)=(-1,1)$

Substitute in the point slope formula.

  $\begin{array}{l}\Rightarrow (y-1)=-1(x-1)\\ \Rightarrow y-1=-x+1\\ \Rightarrow x+y=2\end{array}$

Conclusion:

Therefore, the equation of Jay and Carolyn’s flight to city B is $x+y=2.$

c.

To determine

To find the co-ordinates of city B

Answer to Problem 38PFA

  $(-7.33,9.33)$

Explanation of Solution

Given:

From subpart (b),

Equation of Jay and Carolyn’s flight is $x+y=2$

Equation of Diane’s flight is $y=2x+24$

Calculation:

Consider the equations

  $x+y=\mathrm{2........................}(1)$

  $\begin{array}{l}y=2x+24\\ \Rightarrow 2x-y=-\mathrm{24............}(2)\end{array}$

Add equation (1) and equation (2),

  $\begin{array}{l}\Rightarrow (x+y)+(2x-y)=2-24\\ \Rightarrow 3x=-22\\ \Rightarrow x=-7.33\end{array}$

Substitute the value of x in equation (1),

  $\begin{array}{c}\Rightarrow -7.33+y=2\\ \Rightarrow y=9.33\end{array}$

Conclusion:

Therefore, the coordinates of city B is $(-7.33,9.33).$