# 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?

Jun 15, 2017

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

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

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

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

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

Simplifying
$6 x = 15$

1 Hot dog = $2.50 Substitute the value calculated above into Equation 1 $15 + 3 y = 24$$3 y = 9$1 Nacho =$3.00

Jun 15, 2017

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 Jun 15, 2017 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:
$6 h + 3 n = 24$
$8 h + n = 23$

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

Now, we can subtract the first equation from the second equation to cancel out the $n$ term:
$24 h + 3 n - \left(6 h + 3 n\right) = 69 - 24$
$24 h - 6 h = 45$
$18 h = 45$
$h = \frac{45}{18} = \frac{5}{2} = 2.50$

Finally, we can find $n$ by substituting into either of the original equations and solving. We will use the second equation:
$8 \left(\frac{5}{2}\right) + n = 23$
$20 + n = 23$
$n = 23 - 20$
$n = 3$

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