Question

# The speed of each of the runners by modeling the

Modeling
The speed of each of the runners by modeling the system of linear equations.

2021-08-19

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.
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 $$\displaystyle={\frac{{{d}{i}{s}{\tan{{c}}}{e}}}{{{s}{p}{e}{e}{d}}}}$$
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.
$$\begin{array}{|c|c|} \hline & Distance & Speed & Time \\ \hline First\ runner & 12 & x+2 & t \\Second\ runner & 9 & x & t\\ \hline \end{array}$$
For first runner, use distance $$\displaystyle={s}{p}{e}{e}{d}\times{t}{i}{m}{e}$$ and obtain the equation:
$$\displaystyle{12}={\left({x}+{2}\right)}\times{t}$$
$$\displaystyle{t}={\frac{{{12}}}{{{x}+{2}}}}$$
For second runner, use distance $$\displaystyle={s}{p}{e}{e}{d}\times{t}{i}{m}{e}$$ and obtain the equation:
$$\displaystyle{S}{9}={\left({x}\right)}\times{t}$$
$$\displaystyle{t}={\frac{{{9}}}{{{x}}}}$$
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:
$$\displaystyle{\frac{{{12}}}{{{x}+{2}}}}={\frac{{{9}}}{{{x}}}}$$
$$\displaystyle{12}{x}={9}{\left({x}+{2}\right)}$$
$$\displaystyle{12}{x}={9}{x}+{18}$$
Solve the above equation to obtain the value of variable x as:
$$\displaystyle{12}{x}-{9}{x}={18}$$
$$\displaystyle{3}{x}={18}$$
$$\displaystyle{x}={\frac{{{18}}}{{{3}}}}$$
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.