Determine the values of a for which the system has no solutions, exactly one solution, or infini

Determine the values of a for which the system has no...

Key Concepts

Systems of Linear Equations

Systems of linear equations consist of multiple linear equations that are solved simultaneously for their variable values. Analyzing such systems involves determining whether they are consistent or inconsistent, and whether they have a unique solution or infinitely many solutions. This concept underlies methods for solving equations and understanding the relationships between the variables.

Parametric Systems

Parametric systems include one or more parameters in their coefficients or constant terms, and the values of these parameters can influence the system's behavior. Analyzing a parametric system involves determining how the existence and number of solutions change as the parameter values vary.

Gaussian Elimination

Gaussian elimination is a systematic method used to simplify a system of equations by eliminating variables through row operations. This process transforms the system into a form that makes it possible to easily determine whether a unique solution exists, or if the system has infinitely many solutions or is inconsistent.

Determinant and Unique Solutions

In square systems of linear equations, the determinant of the coefficient matrix is a key factor in determining the nature of the solution. A nonzero determinant implies that the matrix is invertible, which guarantees a unique solution, whereas a zero determinant indicates that the system may either have infinitely many solutions or no solution at all.

Consistency and Inconsistency Conditions

The concepts of consistency and inconsistency refer to whether a system of linear equations has at least one solution or none. A consistent system will either have a unique solution or infinitely many solutions depending on the rank of the coefficient matrix versus the augmented matrix, while an inconsistent system contains contradictory equations and has no solution.

Transcript

00:01 Right, this question gives us a system of equations, and we're trying to figure out what values of a would make that system have no solutions, exactly one solution, or infinitely many.

00:12 So my first step with this, first of all, make it into a matrix with the augmented matrix using the coefficients.

00:23 And once you're there, really what i'm trying to get to is row -reduce a shulun form.

00:29 That's the easiest form from which we can solve a system.

00:35 So that was my goal.

00:36 And i went ahead and did this because i don't necessarily think this is the focus of the question, i guess.

00:44 So i show here what calculations i did.

00:49 So i'm trying to make this into a zero.

00:54 I'm trying to make this into a zero with this first step.

00:57 So i took row 1 multiplied by negative 3 and added to this row for the first calculation and the negative 4 times the first row added to this should be row 3.

01:13 Let me change that with us.

01:17 So maybe that's something perhaps as we go through this that you can kind of pause on if you need to.

01:23 But doing those calculations, you'll did this matrix.

01:26 And at this point, i'm trying to make this into a zero, this negative seven.

01:34 So i noticed from here all i need to do is take the negative of all the numbers in this row, add to row three, and then that would get me where i need to go.

01:44 So this is the calculation i did in this step.

01:50 And this landed me here.

01:52 It's not quite row -reducedural inform, but i have these.

01:58 Zeros in the triangle here.

02:01 And really where the number of solutions is going to come from is from this last row.

02:08 And so that's what i'm going to be working with as we tackle the different types of solutions.

02:17 So to have infinitely many solutions, given that we can get it to this form and everything, really what we're needing for infinitely many is for this to be a zero and this to be a zero.

02:36 The reason why is that we need a free variable.

02:39 We need a parameter that is open or free in our answer.

02:46 So that can be anything, meaning we can have infinitely many answers based on that value.

02:51 So in this case, we need a squared minus 6 .5.

02:59 16 equal to 0 and a minus 4 equal to 0.

03:05 I'll scroll down just a little bit.

03:08 We can solve this pretty simply.

03:12 In order for this to be true, we can square both sides...