At a baseball game Gina purchased 6 hotdogs and 3 nachos for $24. Frank purchased 8 hotdogs and 1 nacho for $23. How much is one hotdog and one nacho?

Let #x# be the cost of 1 hotdog
let #y# be the cost of 1 nacho

#6x + 3y = 24# (Equation 1)

#8x + 1y = 23# (Equation 2)

Divide equation 1 by 3
#2x + y = 8# (Equation 3)

Subtract Equation 3 from Equation 2
#8x - 2x +y - y = 23 - 8#

Simplifying
#6x = 15#

1 Hot dog = $2.50

Substitute the value calculated above into Equation 1
#15 + 3y = 24#
#3y = 9#
1 Nacho = $3.00

Let's call the price of a hotdog: #h#

Let's call the price of a nacho: #n#

We can then write:

#6h + 3n = $24#

#8h + 1n = $23#

Step 1) Solve the second equation for #n#:

#-color(red)(8h) + 8h + 1n = -color(red)(8h) + $23#

#0 + 1n = -8h + $23#

#n = -8h + $23#

Step 2) Substitute #(-8h + $23)# for #n# in the first equation and solve for #h#:

#6h + 3n = $24# becomes:

#6h + 3(-8h + $23) = $24#

#6h + (3 * -8h) + (3 * $23) = $24#

#6h + (-24h) + $69 = $24#

#6h - 24h + $69 = $24#

#(6 - 24)h + $69 = $24#

#-18h + $69 = $24#

#-18h + $69 - color(red)($69) = $24 - color(red)($69)#

#-18h + 0 = $-45#

#-18h = $-45#

#(-18h)/color(red)(-18) = ($-45)/color(red)(-18)#

#(color(red)(cancel(color(black)(-18)))h)/cancel(color(red)(-18)) = $2.50#

#h = $2.50#

Step 3) Substitute #$2.50# for #h# in the solution to the second equation at the end of Step 1 and calculate #n#:

#n = -8h + $23# becomes:

#n = (-8 * $2.50) + $23#

#n = -$20.00 + $23.00#

#n = $3.00#

Hotdogs cost: $2.50

Nachos cost $3.00

Let #h# be the cost of one hotdog and #n# be the cost of one nacho.

We can set up a system of equations:
#6h+3n=24#
#8h+n=23#

We can solve for #h# and #n# by using elimination by multiplying the second equation by #3#:
#24h+3n=69#

Now, we can subtract the first equation from the second equation to cancel out the #n# term:
#24h+3n-(6h+3n)=69-24#
#24h-6h=45#
#18h=45#
#h=45/18=5/2=2.50#

Finally, we can find #n# by substituting into either of the original equations and solving. We will use the second equation:
#8(5/2)+n=23#
#20+n=23#
#n=23-20#
#n=3#

Therefore the cost of one hotdog and one nacho is #h+n=2.5+3=5.5# or #$5.50#