Systems of Linear Equations - Graphical Approach
This interactive HTML5 applet helps you explore and interpret graphically the solutions of a 2×2 system of linear equations of the form:
\( a_1 x + b_1 y = c_1 \)
\( a_2 x + b_2 y = c_2 \)
Use the sliders to change coefficient values \(a_1\), \(b_1\), \(c_1\), \(a_2\), \(b_2\), and \(c_2\) to explore different systems. The graphical interpretation complements algebraic methods like elimination and Cramer's rule. Also check our tutorial on solving systems analytically.
Interactive Tutorial
Instructions: Click "Draw" to start. Default values give the system:
\( x + 3y = 5 \)
\( 2x - 2y = 2 \)
The intersection point \((2, 1)\) solves both equations.
Key Relationship
The intersection point of two lines representing a system's equations is the system's solution.
Exploration Exercises
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Intersection Point Verification
Adjust sliders so lines intersect at a new point \((x, y)\). Verify algebraically (substitution, elimination, or Cramer's rule) that \((x, y)\) solves the system.
Answer
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Parallel Lines Case
Set coefficients so \(a_1 b_2 - a_2 b_1 = 0\) (e.g., \(a_1=2\), \(b_1=4\), \(a_2=1\), \(b_2=2\)).
- How are the lines positioned?
- How many solutions exist?
- If \(a_1 b_2 - a_2 b_1 = 0\), what's the relationship between the lines' slopes?
Answer
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Coincident Lines Case
Set coefficients so \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) (e.g., \(a_1=1\), \(b_1=1\), \(c_1=1\), \(a_2=2\), \(b_2=2\), \(c_2=2\)).
- How many solutions does the system have?
- Explain why.
Answer
Additional Resources