Questions: Solve the system using addition/elimination method. If there is exactly one solution, write as an ordered pair. If not, choose one of the other options. 3x-3y=-21 x+3y=5 One solution: No solution Infinite number of solutions

Solve the system using addition/elimination method. If there is exactly one solution, write as an ordered pair. If not, choose one of the other options.


3x-3y=-21
x+3y=5


One solution: 
No solution
Infinite number of solutions
Transcript text: Solve the system using addition/elimination method. If there is exactly one solution, write as an ordered pair. If not, choose one of the other options. \[ \left\{\begin{array}{l} 3 x-3 y=-21 \\ x+3 y=5 \end{array}\right. \] One solution: $\square$ No solution Infinite number of solutions
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Solution

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Solution Steps

To solve the system of equations using the addition/elimination method, we aim to eliminate one of the variables by adding the equations. First, we can add the two equations directly to eliminate \( y \). If the resulting equation provides a valid solution for \( x \), we substitute back to find \( y \). If the equations result in a contradiction, there is no solution. If they result in a tautology, there are infinite solutions.

Step 1: Write the System of Equations

We start with the given system of equations: \[ \begin{align*}

  1. & \quad 3x - 3y = -21 \\
  2. & \quad x + 3y = 5 \end{align*} \]
Step 2: Solve for One Variable

Using the elimination method, we can manipulate the equations to eliminate one variable. We can multiply the second equation by 3 to align the coefficients of \( y \): \[ 3(x + 3y) = 3(5) \implies 3x + 9y = 15 \]

Step 3: Add the Equations

Now we have: \[ \begin{align*}

  1. & \quad 3x - 3y = -21 \\
  2. & \quad 3x + 9y = 15 \end{align*} \] We can subtract equation 1 from equation 3: \[ (3x + 9y) - (3x - 3y) = 15 - (-21) \implies 12y = 36 \]
Step 4: Solve for \( y \)

Dividing both sides by 12 gives: \[ y = \frac{36}{12} = 3 \]

Step 5: Substitute to Find \( x \)

Now we substitute \( y = 3 \) back into one of the original equations, using equation 2: \[ x + 3(3) = 5 \implies x + 9 = 5 \implies x = 5 - 9 = -4 \]

Final Answer

The solution to the system of equations is: \[ \boxed{(x, y) = (-4, 3)} \]

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