Question

1. olive and pluto are carrying crates and barrels. olive picks up 3 crates and 2 barrels while pluto carries 10 crates and 15 barrels. pluto carried a total of 310 pounds and olive carried 44 pounds.

a. write your let statements
let c = crates lb
let b = barrels lb

b. write two equations, one to show the how much olive carried, and another to show how much pluto carried.
3c + 2b = 44
10c + 15b = 310

c. what is the weight of the crates and what is the weight of the barrels? show all your work.
3c + 2b = 44
-7.5b = 44 - 93

Explanation:

Step1: Identify the system of equations

We have the system:

$$\begin{cases} 3c + 2b = 44 \\ 10c + 15b = 310 \end{cases}$$

Let's simplify the second equation by dividing by 5: \(2c + 3b = 62\). Now we have:

$$\begin{cases} 3c + 2b = 44 \\ 2c + 3b = 62 \end{cases}$$

Step2: Eliminate a variable (e.g., c)

Multiply the first equation by 2: \(6c + 4b = 88\)
Multiply the second equation by 3: \(6c + 9b = 186\)
Subtract the first new equation from the second: \((6c + 9b) - (6c + 4b) = 186 - 88\)
Simplify: \(5b = 98\)? Wait, no, wait, 186 - 88 is 98? Wait, no, 310 divided by 5 is 62, correct. Wait, 442=88, 623=186. 186-88=98? Wait, no, 186-88 is 98? Wait, 186-80=106, 106-8=98. Then 5b=98? That can't be, maybe I made a mistake. Wait, original second equation: 10c +15b=310. Divide by 5: 2c + 3b=62. Correct. First equation: 3c + 2b=44. Let's use elimination for b. Multiply first equation by 3: 9c + 6b=132. Multiply second equation by 2: 4c + 6b=124. Now subtract the second from the first: (9c + 6b) - (4c + 6b)=132 - 124. So 5c=8. Then c=8/5=1.6? Wait, that seems odd. Wait, maybe the initial equations are correct? Wait, Olive: 3 crates and 2 barrels, 44 pounds. Pluto:10 crates and 15 barrels, 310 pounds. Let's check with c=1.6, b=(44 - 31.6)/2=(44 - 4.8)/2=39.2/2=19.6. Then check Pluto: 101.6 +1519.6=16 + 294=310. Oh! Correct. Wait, so crates are 1.6 pounds, barrels 19.6? Wait, that seems unusual, but mathematically it works. Wait, maybe I made a mistake in simplifying. Wait, 10c +15b=310. Let's divide by 5: 2c + 3b=62. Correct. 3c + 2b=44. Let's solve using substitution. From first equation: 3c=44 - 2b → c=(44 - 2b)/3. Substitute into second equation: 2(44 - 2b)/3 + 3b=62. Multiply by 3: 2(44 - 2b) + 9b=186 → 88 - 4b + 9b=186 → 5b=98 → b=19.6. Then c=(44 - 219.6)/3=(44 - 39.2)/3=4.8/3=1.6. So crates are 1.6 pounds, barrels 19.6 pounds.

Answer:

The weight of a crate is \(1.6\) pounds and the weight of a barrel is \(19.6\) pounds.