5. To raise funds for uniforms and travel expenses, the soccer team is holding a car wash in a part of town with a lot of car and truck traffic. The team spent $90 on supplies like sponges and soap. They plan to charge $10 per car and $20 per truck.
Their goal is to raise $460.
How many cars do they have to wash if they washed the following numbers of trucks?
a, 4 trucks
b. 15 trucks
C.21 trucks
d. 27 trucks
e. t trucks
Explain why the equation 2(3x - 5) = 6x + 8 has no solutions.

Let's solve the first problem, where the soccer team is trying to raise $460 and spent $90 on supplies. They charge $10 per car and $20 per truck. To determine how many cars they need to wash for each scenario, we can set up an equation.

### General Approach:
The net amount the team wants to raise is:
\[
\text{Total goal} = 460 - 90 = 370 \, \text{dollars}
\]
So, the team needs to raise **$370** through the car wash after deducting the cost of supplies.

Let:
- $ x $ be the number of cars washed,
- $ t $ be the number of trucks washed.

The money earned from washing $ x $ cars and $ t $ trucks is:
\[
\text{Total revenue} = 10x + 20t
\]
We set this equal to $370:
\[
10x + 20t = 370
\]
This is the general equation that will be used to solve for $ x $ (number of cars) for different values of $ t $ (number of trucks).

### Part (a): 4 trucks

If $ t = 4 $, the equation becomes:
\[
10x + 20(4) = 370
\]
Simplifying:
\[
10x + 80 = 370
\]
Subtract 80 from both sides:
\[
10x = 290
\]
Divide by 10:
\[
x = 29
\]
So, the team must wash **29 cars** if they wash 4 trucks.

### Part (b): 15 trucks

If $ t = 15 $, the equation becomes:
\[
10x + 20(15) = 370
\]
Simplifying:
\[
10x + 300 = 370
\]
Subtract 300 from both sides:
\[
10x = 70
\]
Divide by 10:
\[
x = 7
\]
So, the team must wash **7 cars** if they wash 15 trucks.

### Part (c): 21 trucks

If $ t = 21 $, the equation becomes:
\[
10x + 20(21) = 370
\]
Simplifying:
\[
10x + 420 = 370
\]
Subtract 420 from both sides:
\[
10x = -50
\]
Divide by 10:
\[
x = -5
\]
Since the number of cars cannot be negative, this means that it's not possible to reach their goal by washing 21 trucks alone.

### Part (d): 27 trucks

If $ t = 27 $, the equation becomes:
\[
10x + 20(27) = 370
\]
Simplifying:
\[
10x + 540 = 370
\]
Subtract 540 from both sides:
\[
10x = -170
\]
Divide by 10:
\[
x = -17
\]
Again, since the number of cars cannot be negative, it is impossible to reach their goal by washing 27 trucks alone.

### Part (e): $ t $ trucks (general formula)

For $ t $ trucks, the equation becomes:
\[
10x + 20t = 370
\]
Solve for $ x $:
\[
10x = 370 - 20t
\]
\[
x = \frac{370 - 20t}{10}
\]
\[
x = 37 - 2t
\]
So, the number of cars $ x $ they need to wash if they wash $ t $ trucks is:
\[
x = 37 - 2t
\]
This general equation can be used to find the number of cars for any number of trucks $ t $.

---

### Explaining the equation $ 2(3x - 5) = 6x + 8 $:

1. Start by distributing the left-hand side:
\[
2(3x - 5) = 6x - 10
\]
Now the equation is:
\[
6x - 10 = 6x + 8
\]
2. Subtract $ 6x $ from both sides:
\[
-10 = 8
\]
This is a **contradiction** because $-10$ does not equal $8$. Therefore, the equation has **no solution**. The two sides cannot be equal, no matter what value of $ x $ you choose.

Would you like further clarification or details?

### Five related questions:
1. What would happen if the team raised their prices to $12 per car and $25 per truck?
2. How many trucks would they need to wash if they only washed trucks and no cars?
3. If they had only $70 to spend on supplies, how would that affect the number of cars and trucks to wash?
4. What if the team’s fundraising goal increased to $600 instead of $460?
5. How does changing the number of trucks affect the number of cars in the equation?

### Tip:
Always check for contradictions when solving equations. If both sides don't balance, the equation may have no solution.

To raise funds for uniforms and travel expenses, the soccer team is holding a car wash in a part of town with a lot of car and truck traffic. The team spent $90 on supplies like sponges and soap. They plan to charge $10 per car and $20 per truck. Their goal is to raise $460. How many cars do they have to wash if they washed the following numbers of trucks? a. 4 trucks b. 15 trucks c. 21 trucks d. 27 trucks e. t trucks. Explain why the equation 2(3x - 5) = 6x + 8 has no solutions.

This problem focuses on a soccer team’s car wash fundraiser. Learn how to calculate the number of cars needed to meet their fundraising goal with different numbers of trucks washed. The problem also explains why the equation 2(3x - 5) = 6x + 8 has no solution, introducing students to contradictions in equations. Suitable for Grades 6-8.

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