Answer
$5$ miles of Running, $2$ miles of Swimming, $30$ miles of Cycling.
Work Step by Step
Step 1. Establish the system of equations, assume $x$ miles of Running, $y$ miles of Swimming, and $z$ miles of Cycling :
$\begin{cases} x/10+y/4+z/20=2.5 (hours) \\ x/7.5+y/6+z/15=3 (hours) \\ x/15+y/3+z/40=1.75 (hours) \end{cases}$
Step 2. Remove the fractions of the above equations:
$\begin{cases} 2x+5y+z=50 \\ 4x+5y+2z=90 \\ 8x+40y+3z=210 \end{cases}$
Step 3. Establish the augmented matrix of the system and use the Gauss Eliminations method:
$\begin{vmatrix} 2 & 5 & 1 & 50 \\4 & 5 & 2 & 90\\8 & 40 & 3 & 210 \end{vmatrix} \begin{array}( .\\2R_1-R_2\to R_2\\R_3-4R_1\to R_3\\ \end{array}$
Step 4. Do the operations given on the right side of the matrix.
$\begin{vmatrix} 2 & 5 & 1 & 50 \\0 & 5 & 0 & 10\\0 & 20 & -1 & 10 \end{vmatrix} \begin{array}( .\\.\\.\\ \end{array}$
Step 5. Row 2 gives $5y=10$ and $y=2$mi. Row 3 gives $20y-z=10$ which leads to $z=30$mi. Row 1 gives $2x+5y+z=50$ thus $x=5$mi
Step 6. The final results are: $5$ miles of Running, $2$ miles of Swimming, and $30$ miles of Cycling.
