Questions: Solve the system of linear equations, using the Gauss-Jordan elimination method. (If there is n in terms of the parameters t and/or s.) x-2 y=3 7 x-14 y=21 3 x-6 y=9 (x, y)=(0,-3/2)

Solution Steps

We start with the given system of linear equations:

\[ \begin{align*}

1. & \quad x - 2y = 3 \\
2. & \quad 7x - 14y = 21 \\
3. & \quad 3x - 6y = 9 \end{align*} \]

Notice that the second and third equations can be simplified:

• The second equation \( 7x - 14y = 21 \) can be divided by 7, yielding \( x - 2y = 3 \), which is the same as the first equation.
• The third equation \( 3x - 6y = 9 \) can be divided by 3, yielding \( x - 2y = 3 \) as well.

Thus, all three equations represent the same line, indicating that they are dependent.

Since all three equations are equivalent, we have an infinite number of solutions along the line defined by \( x - 2y = 3 \). We can express \( x \) in terms of \( y \):

\[ x = 2y + 3 \]

To express the solution set, we can let \( y = t \) (where \( t \) is a parameter). Then, substituting \( y \) into the equation for \( x \):

\[ x = 2t + 3 \]

Thus, the solution set can be expressed as:

\[ (x, y) = (2t + 3, t) \]

The system has infinitely many solutions, which can be represented parametrically as:

\[ (x, y) = (2t + 3, t) \quad \text{for any } t \in \mathbb{R} \]

Final Answer