8a) the temperature of one furnace is 1,400°c and cools at a rate of 2.5°c per minute. the temperature of another furnace is 1,200°c and is heated at a rate of 1.5°c per minute. let t represent the time in minutes. write an equation that models when the temperature of the two furnaces will be equal. ______ - ______ t = ______ + ______ t 8b) the temperature of one furnace is 1,400°c and cools at a rate of 2.5°c per minute. the temperature of another furnace is 1,200°c and is heated at a rate of 1.5°c per minute. after how many minutes will the furnaces be the same temperature? ______ 9) sylvester and erin collect baseball cards. sylvester has 250 baseball cards and collects 15 cards per week. erin has 300 baseball cards and collects 5 cards per week. after how many weeks will sylvester and erin have the same number of baseball cards? ______ weeks 10) solve $-\frac{1}{3}(4q - 6) + 2\frac{3}{4} = \frac{2}{3}(q + 1) + 4\frac{1}{4}$. leave your answer as a fraction. q = ______

Question

Accepted Answer

### 8A)
# Explanation:
## Step1: Temperature of first furnace
The first furnace starts at $1400^\circ\text{C}$ and cools at $2.5^\circ\text{C}$ per minute. So its temperature after $t$ minutes is $1400 - 2.5t$.
## Step2: Temperature of second furnace
The second furnace starts at $1200^\circ\text{C}$ and heats at $1.5^\circ\text{C}$ per minute. So its temperature after $t$ minutes is $1200 + 1.5t$.
## Step3: Set temperatures equal
To find when they are equal, set the two expressions equal: $1400 - 2.5t = 1200 + 1.5t$.
# Answer:
$1400 - 2.5t = 1200 + 1.5t$ (So the blanks are $1400$, $2.5$, $1200$, $1.5$)

### 8B)
# Explanation:
## Step1: Start with the equation from 8A
We have $1400 - 2.5t = 1200 + 1.5t$.
## Step2: Add $2.5t$ to both sides
$1400 = 1200 + 1.5t + 2.5t$.
## Step3: Combine like terms
$1400 = 1200 + 4t$.
## Step4: Subtract $1200$ from both sides
$1400 - 1200 = 4t$, so $200 = 4t$.
## Step5: Divide by 4
$t=\frac{200}{4}=50$.
# Answer:
$50$

### 9)
# Explanation:
## Step1: Cards of Sylvester
Sylvester has $250$ cards and collects $15$ per week. So after $w$ weeks, he has $250 + 15w$ cards.
## Step2: Cards of Erin
Erin has $300$ cards and collects $5$ per week. So after $w$ weeks, she has $300 + 5w$ cards.
## Step3: Set equal
Set $250 + 15w = 300 + 5w$.
## Step4: Subtract $5w$ from both sides
$250 + 10w = 300$.
## Step5: Subtract $250$ from both sides
$10w = 50$.
## Step6: Divide by 10
$w = 5$.
# Answer:
$5$

### 10)
# Explanation:
## Step1: Simplify left side
First, distribute $-\frac{1}{3}$ in $-\frac{1}{3}(4q - 6)$: $-\frac{4}{3}q + 2$. Then add $2\frac{3}{4}$: $-\frac{4}{3}q + 2 + 2\frac{3}{4}=-\frac{4}{3}q + 4\frac{3}{4}$.
## Step2: Simplify right side
Distribute $\frac{2}{3}$ in $\frac{2}{3}(q + 1)$: $\frac{2}{3}q + \frac{2}{3}$. Then add $4\frac{1}{4}$: $\frac{2}{3}q + \frac{2}{3}+ 4\frac{1}{4}=\frac{2}{3}q + 4\frac{11}{12}$.
## Step3: Now the equation is
$-\frac{4}{3}q + 4\frac{3}{4}=\frac{2}{3}q + 4\frac{11}{12}$.
## Step4: Subtract $\frac{2}{3}q$ from both sides
$-\frac{4}{3}q-\frac{2}{3}q + 4\frac{3}{4}= 4\frac{11}{12}$, which simplifies to $-2q + 4\frac{3}{4}= 4\frac{11}{12}$.
## Step5: Subtract $4\frac{3}{4}$ from both sides
$-2q=4\frac{11}{12}-4\frac{3}{4}$. Convert to twelfths: $4\frac{11}{12}-4\frac{9}{12}=\frac{2}{12}=\frac{1}{6}$. So $-2q=\frac{1}{6}$.
## Step6: Divide by $-2$
$q=\frac{1}{6}\div(-2)=-\frac{1}{12}$.
# Answer:
$q = -\frac{1}{12}$

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QUESTION IMAGE

Question

Explanation:

8A)

Step1: Temperature of first furnace

Step2: Temperature of second furnace

Step3: Set temperatures equal

Step1: Start with the equation from 8A

Step2: Add \(2.5t\) to both sides

Step3: Combine like terms

Step4: Subtract \(1200\) from both sides

Step5: Divide by 4

Step1: Cards of Sylvester

Step2: Cards of Erin

Step3: Set equal

Step4: Subtract \(5w\) from both sides

Step5: Subtract \(250\) from both sides

Step6: Divide by 10

Answer:

8B)