Find 'a' Values for Ax + Y = 7 & 4X + Ay = 19

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Homework Help Overview

The discussion revolves around finding the values of 'a' for which the system of equations represented by the matrix has no solutions. The equations in question are ax + y = 7 and 4x + ay = 19, which fall under the subject area of linear algebra and systems of equations.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the determinant of the matrix formed by the coefficients of the equations and its role in determining the existence of solutions. There is an exploration of how setting the determinant to zero leads to conditions for 'a'. Some participants express confusion regarding the relationship between the determinant and the solutions.

Discussion Status

The discussion is active, with participants providing hints about the determinant and its implications. There is a recognition that multiple values of 'a' can lead to no solutions, and some guidance has been offered regarding the quadratic equation that arises from the determinant. However, there is no explicit consensus on the correct values of 'a' yet.

Contextual Notes

Participants note that the book provides specific values for 'a' (+/- 2), which raises questions about the derivation of these values from the determinant. There is an ongoing inquiry into the steps needed to arrive at these solutions, indicating a potential gap in understanding the determinant's role.

wayneo
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find the values of 'a' for which the equations { ax + y = 7}
{ 4x + ay = 19}
have no solutions.

I realize that you have to split it into

(a 1) (x) (7)
(4 a) (y) = (19)

but I am stuck how to find solutions for 'a'

any help would be great thanks
 
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Note that this can only happen if the deteminant of your matrix is 0..
 
As arildno said, if the determinant of the matrix[tex]A = \left(<br /> \begin{array}{cc}<br /> a & 1\\<br /> 4 & a<br /> \end{array}<br /> \right)[/tex] is 0 then the equation has no solutions. (Or infinitely many)

Why is this? To solve for x and y you have to multiply both sides by [tex]A^{-1}[/tex]. For [tex]A[/tex] to be invertible, what must be true of the determinant of [tex]A[/tex]?
 
Last edited:
but the answer in the book says +/- 2 how is that
 
'a' can have multiple values; that is, there are multiple matrices for which those equations have no solutions.

Note if you take the determinant of that matix and solve for 'a' you get a quadratic with two solutions.
 
wayneo said:
but the answer in the book says +/- 2 how is that
So far you haven't given any indication that you have understood or tried using the hints given. What is the determinant of that matrix?
What equation for a do you get if you set the determinant equal to 0? What are the solutions to that equation?
 

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