Make up a system of three linear equations containing three...

Make up a system of three linear equations containing three variables that has:
(a) No solution
(b) Exactly one solution
(c) Infinitely many solutions Give the three systems to a friend to solve and critique.

Key Concepts

Infinitely Many Solutions (Dependent Systems)

Infinitely many solutions arise in dependent systems where the equations do not provide completely independent constraints. Instead, one or more of the equations can be expressed as a linear combination of the others, meaning they represent the same geometric object, leading to an overlap of solution sets.

Unique Solution

A unique solution occurs when all the equations in a system intersect at exactly one point, which represents the only set of values that satisfies all equations simultaneously. This usually means that the equations are independent and the system’s coefficient matrix is non-singular.

No Solution (Inconsistent Systems)

A system has no solution when its equations represent parallel geometric objects, such as planes that never intersect. In this case, the conditions set by the equations are contradictory, making it impossible to find any combination of variable values that satisfies all the equations at once.

System of Linear Equations

A system of linear equations is a collection of equations involving the same set of variables. The goal is to find values for these variables that satisfy every equation simultaneously. Systems with three variables usually represent three-dimensional conditions where each equation corresponds to a plane in space.

Consistency and Inconsistency

A system is called consistent if there exists at least one set of values that satisfies all the equations, and inconsistent if no such set exists. Consistency is crucial in determining whether a system has a unique solution or infinitely many solutions.

Transcript

00:01 Okay, so we're going to create systems of three variables for each of the following situation.

00:07 No solution, exactly one solution, and infinite solution.

00:10 All you need to remember is to do a no solution one.

00:14 When you create your three equations, the easiest way to do it is to make sure at least two of the equations are identical in the coefficient values themselves, and just the answers are different.

00:27 So what i mean is if i have x, y, z, then.

00:31 I can have 2x plus 2y plus 2y, plus 2z.

00:36 And all that i have to make sure is that if this answer is 5, this answer is not doubled.

00:42 So this answer could also be 5.

00:45 So same idea with the coefficients, different constants is your answer.

00:50 And then your third one, we could just keep going with the same pattern just to be safe.

00:55 So 3x plus 3y plus 3z and equal 2.

01:02 That way, when we do go to cancel things out, we end up with a false statement where the left side is equal to zero or the right side is equal to a value.

01:12 If we're doing exactly one solution, we're just going to have to create unique equations.

01:17 So we're just going to create three unique equations that are not copies of each other in any way, shape, or four.

01:24 So if i have like x, y, i could have x minus y minus d for one.

01:30 And the answers could all be the same if i wanted to.

01:33 Totally doesn't matter, but we can throw in different answers for fun...

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