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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)

5 answers

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 \(\boxed{(2, 3)}\).

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

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.

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

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