What is the solution to this system of equation in simplified ordered pair form?

6c-9d=11
5c-9d=103

1 answer

To solve the system of equations:

  1. \( 6c - 9d = 11 \) (Equation 1)
  2. \( 5c - 9d = 103 \) (Equation 2)

We can start by eliminating \( d \) from the equations. Since both equations have \(-9d\) as a term, we can subtract Equation 1 from Equation 2.

Subtract Equation 1 from Equation 2:

\[ (5c - 9d) - (6c - 9d) = 103 - 11 \]

This simplifies to:

\[ 5c - 6c + 9d - 9d = 92 \] \[ -c = 92 \]

Thus, we find:

\[ c = -92 \]

Now that we have \( c \), we can substitute \( c = -92 \) back into one of the original equations to find \( d \). We'll use Equation 1:

\[ 6(-92) - 9d = 11 \]

Calculating \( 6(-92) \):

\[ -552 - 9d = 11 \]

Now, isolate \( d \):

\[ -9d = 11 + 552 \] \[ -9d = 563 \]

Dividing both sides by -9 gives:

\[ d = -\frac{563}{9} \]

Now we have \( c \) and \( d \):

\[ c = -92, \quad d = -\frac{563}{9} \]

Thus, the solution to the system of equations in simplified ordered pair form is:

\[ \boxed{\left( -92, -\frac{563}{9} \right)} \]