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6x−3y=−6

9x−3y=−18

1 answer

To solve the system of equations:

  1. \( 6x - 3y = -6 \) (Equation 1)
  2. \( 9x - 3y = -18 \) (Equation 2)

We can start by observing that both equations can be simplified.

Simplifying Equation 1:

\[ 6x - 3y = -6 \] Divide the entire equation by 3: \[ 2x - y = -2 \quad \text{(Equation 1 simplified)} \]

Simplifying Equation 2:

\[ 9x - 3y = -18 \] Divide the entire equation by 3: \[ 3x - y = -6 \quad \text{(Equation 2 simplified)} \]

Now, we have a new system of equations:

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

Solving the System:

Now we can set the two simplified equations equal to each other since they both equal \(y\): \[ 2x + 2 = 3x + 6 \]

Rearranging gives us: \[ 2 = 3x - 2x + 6 \] \[ 2 = x + 6 \] Subtracting 6 from both sides, we find: \[ x = -4 \]

Now we can substitute \(x = -4\) back into one of the simplified equations to find \(y\). We'll use Equation 1 simplified: \[ 2(-4) - y = -2 \] \[ -8 - y = -2 \] Adding 8 to both sides: \[ -y = 6 \] Multiplying by -1 gives us: \[ y = -6 \]

Solution:

The solution to the system of equations is: \[ (x, y) = (-4, -6) \]