avissidep

2021-08-18

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

Asma Vang

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 $=\frac{dis\mathrm{tan}ce}{speed}$
Step 2:
There are two unknowns in the problem. Consider x be the speed of first runner and hence, from the given information, $x+2$ be the speed of second runner.
Step 3:
The faster runner runs with the speed of $x+2$ 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.

For first runner, use distance $=speed×time$ and obtain the equation:
$12=\left(x+2\right)×t$
$t=\frac{12}{x+2}$
For second runner, use distance $=speed×time$ and obtain the equation:
$S9=\left(x\right)×t$
$t=\frac{9}{x}$
Thus, the system of linear equations is:
$t=\frac{12}{x+2}\left(1\right)$
$t=\frac{9}{x}\left(2\right)$
Step 4:
Substitute equation (1) in equation (2).
$\frac{12}{x+2}=\frac{9}{x}$
Thus, solve the equations to find the variable x:
$\frac{12}{x+2}=\frac{9}{x}$
$12x=9\left(x+2\right)$
$12x=9x+18$
Solve the above equation to obtain the value of variable x as:
$12x-9x=18$
$3x=18$
$x=\frac{18}{3}$
S $x=6$
Thus, the value of x is 6miles/hour.
Therefore, the speed of second runner is $x=6$ miles/hour.
S Therefore, the speed of first runner is,
$x+2=6+2=8$ miles/hour
Step 5:
Thus, the speed of first runner is 8 miles/hour and the speed of second runner is 6 miles/hour.

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