Question

1. moe, larry, and curly went to del taco. moe bought three tacos, two burritos, and one nacho and paid $9.50. larry bought two tacos, two burritos, and two nachos and paid $9.00. curly bought six tacos, four burritos and three nachos and paid $20.00. what is the individual cost of each item?
2. the sum of the measures of the three angles in any triangle is 180 degrees. in triangle abc the measure of angle a is twice the sum of the measures of angles b and c. the measure of angle b is three times the measure of angle c. what are the measures of the three angles?
3. at edwards theatre last night, 155 tickets were sold for the new movie. adult tickets sold for $12, student tickets $8, and seniors $9, bringing in $1580 total. the number of adult tickets sold was four times the number of senior tickets. how many of each type of ticket were sold?
4. yoda, anakin, and obi - wan are combining their savings for a flight to the death star. together, they have $680. anakin has saved $20 less than obi - wan. yoda has saved triple what the other two have saved put together. how much has each contributed to the total savings?

Explanation:

Question 2

Step1: Define variables

Let $t$ be the cost of a taco, $b$ be the cost of a burrito, and $n$ be the cost of a nacho. Then we have the following system of equations based on the purchases:
$3t + 2b + n=9.5$ (Moe's purchase)
$2t + 2b+2n = 9$ (Larry's purchase)
$6t + 4b+3n = 20$ (Curly's purchase)

Step2: Simplify the second - equation

Divide the second equation $2t + 2b+2n = 9$ by 2, we get $t + b + n=4.5$. Then $n=4.5 - t - b$.

Step3: Substitute $n$ into the first equation

Substitute $n = 4.5 - t - b$ into $3t + 2b + n=9.5$.
$3t + 2b+(4.5 - t - b)=9.5$
$3t + 2b + 4.5 - t - b=9.5$
$2t + b=5$, so $b = 5 - 2t$.

Step4: Substitute $n$ and $b$ into the third equation

Substitute $n = 4.5 - t - b$ and $b = 5 - 2t$ into $6t + 4b+3n = 20$.
First, $n=4.5 - t-(5 - 2t)=t - 0.5$.
Then $6t+4(5 - 2t)+3(t - 0.5)=20$
$6t + 20-8t+3t - 1.5 = 20$
$(6t-8t + 3t)+(20 - 1.5)=20$
$t+18.5 = 20$
$t = 1.5$

Step5: Find $b$

Substitute $t = 1.5$ into $b = 5 - 2t$, $b=5-2\times1.5=2$.

Step6: Find $n$

Substitute $t = 1.5$ and $b = 2$ into $n=t - 0.5$, $n=1.5 - 0.5 = 1$.

Step1: Define variables

Let $A$, $B$, and $C$ be the measures of angles $A$, $B$, and $C$ respectively. We know that $A + B + C=180$ (sum of angles in a triangle). Also, $A = 2(B + C)$ and $B = 3C$.

Step2: Substitute $A$ in the first - equation

Since $A = 2(B + C)$, then $A + B + C=180$ becomes $2(B + C)+B + C=180$, so $3(B + C)=180$, and $B + C = 60$.

Step3: Substitute $B$ in $B + C = 60$

Substitute $B = 3C$ into $B + C = 60$, we get $3C+C=60$, $4C=60$, $C = 15$.

Step4: Find $B$

Since $B = 3C$, then $B=3\times15 = 45$.

Step5: Find $A$

Since $A + B + C=180$, then $A=180-(B + C)=180 - 60=120$.

Step1: Define variables

Let $a$ be the number of adult tickets, $s$ be the number of student tickets, and $r$ be the number of senior tickets.
We have the following system of equations:
$a + s + r=155$ (total number of tickets)
$12a + 8s+9r = 1580$ (total money)
$a = 4r$ (relationship between adult and senior tickets)

Step2: Substitute $a = 4r$ into the first two equations

The first equation becomes $4r + s + r=155$, so $s=155 - 5r$.
The second equation becomes $12\times4r+8s + 9r=1580$, $48r+8s + 9r=1580$, $57r+8s=1580$.

Step3: Substitute $s = 155 - 5r$ into $57r+8s=1580$

$57r+8(155 - 5r)=1580$
$57r+1240-40r=1580$
$17r=1580 - 1240$
$17r = 340$
$r = 20$

Step4: Find $a$

Since $a = 4r$, then $a=4\times20 = 80$.

Step5: Find $s$

Since $s=155 - 5r$, then $s=155-5\times20=155 - 100 = 55$.

Answer:

The cost of a taco is $\$1.50$, the cost of a burrito is $\$2.00$, and the cost of a nacho is $\$1.00$.

Question 3