Question

Solve the system using multiplication for the linear combination method.

6x – 3y = 3

–2x + 6y = 14

What is the solution to the system?

(2, 1)
(2, –3)
(2, –1)
(2, 3)

Answer 1

Monica’s school band held a car wash to raise money for a trip to a parade in New York City. After washing 125 cars, they made $775 from a combination of $5.00 quick washes and $8.00 premium washes.

Let x represent the number of quick washes and let y represent the number of premium washes. Which system of linear equations represents the situation?

5x + 8y = 775 and x + y =125
5x – 8y = 125 and x + y = 775
5x + 8y = 775 and x – y = 125
5x – 8y = 125 and x – y = 775

Answer 2

Monica’s school band held a car wash to raise money for a trip to a parade in New York City. After washing 125 cars, they made $775 from a combination of $5.00 quick washes and $8.00 premium washes.

This system of equations models the situation.

x + y =125

5x + 8y = 775

Solve the system to answer the questions.

How many premium car washes were ordered?

premium car washes



How many quick car washes were ordered?

quick car washes

Answer 3

To solve the system of equations using the linear combination method, we first start with the given equations:

1. \( 6x - 3y = 3 \) (Equation 1)
2. \( -2x + 6y = 14 \) (Equation 2)

### Step 1: Make the coefficients of \( y \) the same

We can multiply Equation 1 by 2 in order to make the coefficients of \( y \) match (specifically to make them both 6):

\[
2(6x - 3y) = 2(3)
\]
This results in:

\[
12x - 6y = 6 \quad \text{(New Equation 1)}
\]

Now we have:

1. \( 12x - 6y = 6 \) (New Equation 1)
2. \( -2x + 6y = 14 \) (Equation 2)

### Step 2: Add the two equations

Now, we can add the two equations together:

\[
(12x - 6y) + (-2x + 6y) = 6 + 14
\]

This simplifies to:

\[
10x = 20
\]

### Step 3: Solve for \( x \)

Now, divide both sides by 10:

\[
x = 2
\]

### Step 4: Substitute \( x \) back into one of the original equations

We'll substitute \( x = 2 \) back into Equation 1:

\[
6(2) - 3y = 3
\]

This simplifies to:

\[
12 - 3y = 3
\]

### Step 5: Solve for \( y \)

Now, subtract 12 from both sides:

\[
-3y = 3 - 12
\]

This gives us:

\[
-3y = -9
\]

Now divide by -3:

\[
y = 3
\]

### Final Solution:

The solution to the system is \((2, 3)\).

Among the given options:
- (2, 1)
- (2, –3)
- (2, –1)
- **(2, 3)**

The answer is \(\boxed{(2, 3)}\).

Answer 4

To set up the system of linear equations representing the situation described, we need to consider the two pieces of information given:

1. The total number of cars washed is 125.
2. The total amount of money made from the washes is $775, where quick washes cost $5.00 each and premium washes cost $8.00 each.

Let:
- \( x \) be the number of quick washes.
- \( y \) be the number of premium washes.

### Step 1: Create the first equation based on the total number of cars washed.

The total number of cars washed can be represented as:
\[
x + y = 125 \quad \text{(Equation 1)}
\]

### Step 2: Create the second equation based on the total money earned.

The total amount of money made can be represented as:
\[
5x + 8y = 775 \quad \text{(Equation 2)}
\]

### Conclusion

The system of linear equations that represents this situation is:

\[
\begin{cases}
5x + 8y = 775 \\
x + y = 125
\end{cases}
\]

Thus, the correct answer is:

**5x + 8y = 775 and x + y = 125.**

Answer 5

To solve the system of equations you provided, we have:

1. \( x + y = 125 \) (Equation 1)
2. \( 5x + 8y = 775 \) (Equation 2)

Where:
- \( x \) is the number of quick washes.
- \( y \) is the number of premium washes.

### Step 1: Solve Equation 1 for \( y \)

From Equation 1, we can express \( y \) in terms of \( x \):

\[
y = 125 - x \quad \text{(Equation 3)}
\]

### Step 2: Substitute Equation 3 into Equation 2

Now, substitute Equation 3 into Equation 2:

\[
5x + 8(125 - x) = 775
\]

### Step 3: Simplify and solve for \( x \)

Now distribute the \( 8 \):

\[
5x + 1000 - 8x = 775
\]

Combine like terms:

\[
-3x + 1000 = 775
\]

Subtract \( 1000 \) from both sides:

\[
-3x = 775 - 1000
\]

This gives:

\[
-3x = -225
\]

Now divide by \(-3\):

\[
x = 75
\]

### Step 4: Substitute \( x \) back into Equation 3 to find \( y \)

Now, use Equation 3 to find \( y \):

\[
y = 125 - 75
\]
\[
y = 50
\]

### Conclusion:

So, the solutions are:
- The number of quick car washes (ordered) is \( x = 75 \).
- The number of premium car washes (ordered) is \( y = 50 \).

To summarize:
- **Premium car washes:** 50
- **Quick car washes:** 75