Geometry of Linear Systems in 2D

Row Picture (Lines)

When you click Solve System, you see the row picture:

• Each equation represents a line in 2D space
• \(2x + y = 8\) is line \(L_1\)
• \(4x - 3y = 6\) is line \(L_2\)
• The solution is where the lines intersect

Column Picture (Vectors)

When you click Show Linear Combination, you see the column picture:

• The system becomes: \(c_1 \mathbf{v_1} + c_2 \mathbf{v_2} = \mathbf{b}\)
• \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are the column vectors
• We find scalars \(c_1, c_2\) that combine them to reach \(\mathbf{b}\)
• The parallelogram shows the vector addition

How They Connect

The inputs are synchronized. For the system:

\[\begin{cases} 2x + y = 8 \\ 4x - 3y = 6 \end{cases}\]

Row form: Two equations, two unknowns

Column form:

\[x \begin{bmatrix} 2 \\ 4 \end{bmatrix} + y \begin{bmatrix} 1 \\ -3 \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \end{bmatrix}\]

The solution \(x=3, y=2\) becomes \(c_1=3, c_2=2\) in the vector view.

Try It

• Change the augmented matrix to see different line configurations
• Modify \(c_1, c_2\) to explore different linear combinations
• Zoom and pan to examine the solution closely
• Parallel lines = no solution, coincident lines = infinite solutions