Answer:
300 dimes and 120 quarters
Step-by-step explanation:
Let x be the number of dimes and y be the number of quarters Travis has. Travis has $60 (6000 cents) in dimes and quarters, then
[tex]10x+25y=6000.[/tex]
If he could switch the numbers of dimes with the number of quarter, he would have y dimes and x quarters. In total this amount of money is $87 (8700 cents), then
[tex]10y+25x=8700.[/tex]
Solve the system of two equations:
[tex]\left\{\begin{array}{l}10x+25y=6000\\ \\10y+25x=8700\end{array}\right.\Rightarrow \left\{\begin{array}{l}2x+5y=1200\\ \\2y+5x=1740\end{array}\right..[/tex]
Multiply the 1st equation by 5, the 2nd equation by 2 and subtract them:
[tex]\left\{\begin{array}{l}10x+25y=6000\\ \\10x+4y=3480\end{array}\right.\Rightarrow 21y=2520,\ y=120.[/tex]
Then
[tex]2x+5\cdot 120=1200,\\ \\2x=1200-600,\\ \\2x=600,\\ \\x=300.[/tex]
