Question

based on the equation $t + 2p+2s = 2t + 3s$, determine if these equations are true or false.
(a) $t + 2p + s=2t + 2s$
(b) $3t + 2p = 5s$
(c) $2p + 2s=t + 3s$
(d) $t + 2p=2t + s$
(e) $2p + 5s = 3t$
(f) $2p=t + s$

Explanation:

Step1: Simplify the given equation

Starting with $t + 2p+2s=2t + 3s$, we can get $2p=t + s$ by subtracting $t$ and $2s$ from both sides.

Step2: Check equation (a)

Given $t + 2p + s=2t + 2s$. Substitute $2p=t + s$ into it, we have $t+(t + s)+s=2t + 2s$, which simplifies to $2t+2s=2t + 2s$. So (a) is true.

Step3: Check equation (b)

Given $3t + 2p=5s$. Substitute $2p=t + s$ into it, we get $3t+(t + s)=5s$, which simplifies to $4t + s=5s$ or $4t = 4s$ or $t = s$. But from the original $2p=t + s$ we can't be sure $t = s$. So (b) is false.

Step4: Check equation (c)

Given $2p+2s=t + 3s$. Substitute $2p=t + s$ into it, we have $(t + s)+2s=t + 3s$, which simplifies to $t + 3s=t + 3s$. So (c) is true.

Step5: Check equation (d)

Given $t + 2p=2t + s$. Substitute $2p=t + s$ into it, we get $t+(t + s)=2t + s$, which simplifies to $2t + s=2t + s$. So (d) is true.

Step6: Check equation (e)

Given $2p+5s=3t$. Substitute $2p=t + s$ into it, we have $(t + s)+5s=3t$, which simplifies to $t+6s=3t$ or $2t = 6s$ or $t = 3s$. But from the original $2p=t + s$ we can't be sure $t = 3s$. So (e) is false.

Step7: Check equation (f)

Given $2p=t + s$. This is the same as the result we got from simplifying the original equation. So (f) is true.

Answer:

a. True
b. False
c. True
d. True
e. False
f. True