Question

What is the solution to this system of equation in simplified ordered pair form using substutition?
6c-9d=111
5c-9d=103

Answer 1

To solve the system of equations using substitution, we start with the following equations:

1. \( 6c - 9d = 111 \) (Equation 1)
2. \( 5c - 9d = 103 \) (Equation 2)

First, we'll isolate one of the variables in one of the equations. Let's isolate \(d\) in terms of \(c\) using Equation 2.

From Equation 2:

\[
5c - 9d = 103
\]

Rearranging gives:

\[
9d = 5c - 103
\]
\[
d = \frac{5c - 103}{9} \quad (Equation 3)
\]

Now that we have \(d\) in terms of \(c\), we can substitute this expression for \(d\) into Equation 1:

From Equation 1:

\[
6c - 9d = 111
\]

Substituting for \(d\) from Equation 3:

\[
6c - 9\left(\frac{5c - 103}{9}\right) = 111
\]

The \(9\) in the denominator and the \(9\) in front cancel out:

\[
6c - (5c - 103) = 111
\]

This simplifies to:

\[
6c - 5c + 103 = 111
\]
\[
c + 103 = 111
\]

Next, we isolate \(c\):

\[
c = 111 - 103
\]
\[
c = 8
\]

Now that we have the value of \(c\), we can substitute it back into Equation 3 to find \(d\):

\[
d = \frac{5(8) - 103}{9}
\]
\[
d = \frac{40 - 103}{9}
\]
\[
d = \frac{-63}{9}
\]
\[
d = -7
\]

Thus, we have found \(c\) and \(d\):

\[
c = 8 \quad \text{and} \quad d = -7
\]

Finally, we present the solution in ordered pair form:

\[
(c, d) = (8, -7)
\]

Therefore, the solution to the system of equations is:

\[
\boxed{(8, -7)}
\]