The speed of each of the runners by modeling the

Procedure used:
"odel and solve system of linear equations
1) Identify the category of given problem.
2) Recognize and algebraically name the unknowns.
3) Translate the problem statement as a system of linear equations using the variables from Step 2.
S 4) Solve the system as obtained in Step 3.
5) State the answer statemen."
Calculation:
Step 1:
The given problem is a uniform motion problem where speed of the runner is to be calculated using the following formula:
time ${\displaystyle =\frac{dis\tan ce}{speed}}$
Step 2:
There are two unknowns in the problem. Consider x be the speed of first runner and hence, from the given information, ${\displaystyle x+2}$ be the speed of second runner.
Step 3:
The faster runner runs with the speed of ${\displaystyle x+2}$ with the distance of 12 miles and second runner runs with x speed and the distance covered is 9 miles. It is summarized in Table.
$\begin{array}{|cccc|}\hline & Distance& Speed& Time\\ Firstrunner& 12& x+2& t\\ Secondrunner& 9& x& t\\ \hline\end{array}$
For first runner, use distance ${\displaystyle =speed\times time}$ and obtain the equation:
${\displaystyle 12=(x+2)\times t}$
${\displaystyle t=\frac{12}{x+2}}$
For second runner, use distance ${\displaystyle =speed\times time}$ and obtain the equation:
${\displaystyle S9=\left(x\right)\times t}$
${\displaystyle t=\frac{9}{x}}$
Thus, the system of linear equations is:
${\displaystyle t=\frac{12}{x+2}\left(1\right)}$
${\displaystyle t=\frac{9}{x}\left(2\right)}$
Step 4:
Substitute equation (1) in equation (2).
${\displaystyle \frac{12}{x+2}=\frac{9}{x}}$
Thus, solve the equations to find the variable x:
${\displaystyle \frac{12}{x+2}=\frac{9}{x}}$
${\displaystyle 12x=9(x+2)}$
${\displaystyle 12x=9x+18}$
Solve the above equation to obtain the value of variable x as:
${\displaystyle 12x-9x=18}$
${\displaystyle 3x=18}$
${\displaystyle x=\frac{18}{3}}$
S ${\displaystyle x=6}$
Thus, the value of x is 6miles/hour.
Therefore, the speed of second runner is ${\displaystyle x=6}$ miles/hour.
S Therefore, the speed of first runner is,
${\displaystyle x+2=6+2=8}$ miles/hour
Step 5:
Thus, the speed of first runner is 8 miles/hour and the speed of second runner is 6 miles/hour.