Use the substitution method to solve the system 2x + 2y = 6,3x + 2y = 9.

1 answer

To solve the system of equations using the substitution method, we first take one equation and solve for one variable in terms of the other.

We have the following system:

  1. \( 2x + 2y = 6 \)
  2. \( 3x + 2y = 9 \)

Let's start with the first equation and solve for \( y \):

  1. Rearrange the first equation:

\[ 2y = 6 - 2x \]

  1. Divide by 2:

\[ y = 3 - x \]

Now we have expressed \( y \) in terms of \( x \). Next, we substitute this expression for \( y \) into the second equation:

  1. Substitute \( y = 3 - x \) into the second equation \( 3x + 2y = 9 \):

\[ 3x + 2(3 - x) = 9 \]

  1. Distribute the 2:

\[ 3x + 6 - 2x = 9 \]

  1. Combine like terms:

\[ x + 6 = 9 \]

  1. Subtract 6 from both sides:

\[ x = 3 \]

Now that we have \( x = 3 \), we can substitute this value back into the expression for \( y \):

\[ y = 3 - x = 3 - 3 = 0 \]

So, the solution to the system of equations is:

\[ x = 3, \quad y = 0 \]

Thus, the solution (in ordered pair form) is:

\[ \boxed{(3, 0)} \]