Question

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?

Answer 1

1 Hot dog = $2.50; 1 Nacho = $3.00

Explanation:

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

Answer 2

See a solution process below:

Explanation:

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

Answer 3

Answer: `$5.50`

Explanation:

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`