Question

Fred walked a trail at 2 mph. Paul started 30 minutes later on the same trail and walked at a pace of 3 mph. Create a system of equations in which y represents the distance walked in miles and x represents the time walked in hours. Explain what this solution means in terms of Paul catching up to Fred.

Answer 1

Answer:

$y=2x\\y=3(x-\frac{1}{2})$

Paul will catch Fred after $1\ hours$ at a distance $3\ miles$.

Step-by-step explanation:

Distance is represented by $y$ and the time is represented by $x$. Here after $x$ hours(when Fred started) the distance covered by Fred and Paul is same that is $y$.

For Fred:

time=$x$

distance=$y$

Speed$=2\ mph$

$Distance=speed\times time\\y=2\times x \\y=2x$

Paul:

Since Fred started 30 minutes earlier, so Paul's travel time will be 30 minutes less than that of Fred.

Time$=x-\frac{1}{2}$

Speed$=3\ mph$

Distance$=y$

$y=3\times (x-\frac{1}{2})\\y=3(x-\frac{1}{2})$

Hence system of equation

$y=2x\\y=3(x-\frac{1}{2})$

Solution of this system of equations gives the time, when Paul will catch Fred and distance covered by each of them.

$y=2x\\y=3(x-\frac{1}{2})\\\\2x=3x-\frac{3}{2}\\x=\frac{3}{2}\\\\y=2x\\y=2\times \frac{3}{2}\\y=3$

Time taken by Paul$=x-\frac{1}{2}=\frac{3}{2}-\frac{1}{2}=1$

Hence Paul will catch Fred after $1\ hours$ at a distance $3\ miles$.