First:

Let's call the number of quarters: #q#

Let's call the number of nickles: #n#

We know:

#q + n = 65# because there are 65 coins in the jar.

We also know multiplying the value of the coins by the number of coins gives: #0.25q + 0.05n = $10.25#

Step 1) We can now solve the first equation for #q#:

#q + n = 65#

#q + n - color(red)(n) = 65 - color(red)(n)#

#q + 0 = 65 - n#

#q = 65 - n#

Step 2) Substitute #(65 - n)# for #q# in the second equation and solve for #n#:

#0.25q + 0.05n = 10.25# becomes:

#0.25(65 - n) + 0.05n = 10.25#

#(0.25 * 65) - (0.25 * n) + 0.05n = 10.25#

#16.25 - 0.25n + 0.05n = 10.25#

#16.25 + (-0.25 + 0.05)n = 10.25#

#16.25 + (-0.2)n = 10.25#

#16.25 - 0.2n = 10.25#

#-color(red)(16.25) + 16.25 - 0.2n = -color(red)(16.25) + 10.25#

#0 - 0.2n = -6#

#-0.2n = -6#

#(-0.2n)/color(red)(-0.2) = -6/color(red)(-0.2)#

#(color(red)(cancel(color(black)(-0.2)))n)/cancel(color(red)(-0.2)) = 30#

#n = 30#

Step 3) Substitute #30# for #n# in the solution to the first equation at the end of Step 1 and calculate #q#:

#q = 65 - n# becomes:

#q = 65 - 30#

#q = 35#

The solution is: #n = 30# and #q = 35#

So there are #30# nickles and #35# quarters.