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1х 3х + 3y =15

8x + 5y = 46
What is the solution to the system of equations above?

x A x=-3, y = 6

ax B x =7,y=-2

C
X=6, y=-4

x D x=-2, y= -6

3 answers

To solve the system of equations given:

  1. \( 3x + 3y = 15 \)
  2. \( 8x + 5y = 46 \)

We can simplify the first equation:

  1. \( 3x + 3y = 15 \) can be simplified by dividing everything by 3:

\[ x + y = 5 \quad (1) \]

Now we have a simpler equation (1) and the second equation remains the same (2):

  1. \( 8x + 5y = 46 \)

Now we can express \( y \) from equation (1):

\[ y = 5 - x \quad (3) \]

Next, we substitute equation (3) into equation (2):

\[ 8x + 5(5 - x) = 46 \]

Expanding:

\[ 8x + 25 - 5x = 46 \]

Now, combine like terms:

\[ 3x + 25 = 46 \]

Next, subtract 25 from both sides:

\[ 3x = 21 \]

Now, divide by 3:

\[ x = 7 \]

Now that we have \( x \), we can find \( y \) using equation (3):

\[ y = 5 - x = 5 - 7 = -2 \]

Thus, the solution to the system of equations is \( x = 7 \) and \( y = -2 \).

Looking at the options provided, the correct answer is:

B \( x = 7, y = -2 \)

Ax A group of 16 people bought tickets to a museum. They paid a total of $56. Each adult ticket costs $4 and each child ticket costs $2. Formulate and solve a system of equations to determine how many adult tickets and child tickets did they buy.

A
4 adult tickets, 12 child tickets

8 adult tickets, 8 child tickets

C
10 adult tickets, 6 child tickets

D
12 adult tickets, 4 child tickets

To solve the problem, we need to create a system of equations based on the information provided:

Let:

  • \( x \) = number of adult tickets
  • \( y \) = number of child tickets

From the problem, we can derive the following two equations:

  1. Total Number of People: \[ x + y = 16 \quad (1) \]

  2. Total Amount Paid: \[ 4x + 2y = 56 \quad (2) \]

Now, we can solve these equations step by step.

Step 1: Solve equation (1) for \( y \)

From equation (1): \[ y = 16 - x \quad (3) \]

Step 2: Substitute \( y \) in equation (2)

Now substitute equation (3) into equation (2): \[ 4x + 2(16 - x) = 56 \]

Expanding gives: \[ 4x + 32 - 2x = 56 \]

Combining like terms yields: \[ 2x + 32 = 56 \]

Step 3: Solve for \( x \)

Now subtract 32 from both sides: \[ 2x = 24 \] Dividing by 2 gives: \[ x = 12 \]

Step 4: Substitute \( x \) back to find \( y \)

Substituting \( x \) back into equation (3): \[ y = 16 - 12 = 4 \]

Solution

Thus, the solution is:

  • Adult tickets: \( x = 12 \)
  • Child tickets: \( y = 4 \)

So the group bought 12 adult tickets and 4 child tickets.

Looking at the options provided, the correct answer is:

D \( 12 \) adult tickets, \( 4 \) child tickets.