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Exploration Guide: Systems of Linear Equations (Classic) Gizmo
Systems of Linear Equations (Classic)
Adjust the numerical values into the equations and observe the effect on the graphs of the two lines.
Adjust the numerical values until the two lines intersect and record the point of intersection (mouse over the point to see its coordinates).
What happens if you multiply the first equation by 2? (Double each coefficient and the right-hand-side.) Explain.
Predict what will happen to the graph and the intersection point if we replace the second equation with the sum of the two equations. Try it.
Make some conjectures based on what you have observed.
Create a system of equations that has no solution (no intersection point).
What do you observe about the slopes of the two lines?
Create a system of equations that has infinitely many solutions.
What is different about the two lines in this case as compared to the "no solution" case?
Turn on the "show determinant" option and observe the value of the determinant as you create systems with one solution, no solutions, or infinitely many solutions.
Conjecture how you might use the determinant to give you information on the number of solutions a linear system has.
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