# -5x-4y=2a 4x-5y=2 which of the following choices of a will result in a system of linear equations with exactly one solution?

A. a can be any number

B. a can be any number except .8

C. a can be any number but -.8

D. a equals .8

Relevance
• - 5x -  4y =  2a, ie., 4y = -5x -2a, ie., y = -(5/4)x -(1/2)...(1).;

4x -  5y =  2... ie., 5y =   4x - 2, ie.,  y =  (4/5)x -(2/5)...(2).;

Clearly, both lines defined in (1) & (2) are not //, guaranteed to intersect at a single

point irregardless of the value of a.

• The coefficients on x and y are the numbers that will determine whether there's a unique solution or not.  In the system:

b₁x + c₁y = d₁

b₂x + c₂y = d₂

...there's a unique solution whenever D = (b₁c₂ - b₂c₁) is nonzero.  In your problem that's:

D = (-5)(-5) - (-4)(4) = 25 + 16 = 41

So, there's exactly one solution to your system no matter what the constant terms (2a and 2) are, and any value of a will work.

That expression for D is the determinant of the 2x2 matrix made from the coefficients on x and y.  When the determinant of a matrix is nonzero, that means the rows of the matrix are independent.  No row can be made by adding multiples of the other rows.

It's only when the determinant is zero that you can have multiple solutions (two lines coinciding in the 2-variable case) or no solution (two parallel lines that never intersect).

• In order for there to be one solution, the slopes have to be different.  Let's put both equations into slope-intercept form:

-5x - 4y = 2a and 4x - 5y = 2

-4y = 5x + 2a and -5y = -4x + 2

y = (-5/4)x - (1/2)a and y = (4/5)x - (2/5)

Since the slopes are different no matter what "a" is, "a" can be any value and you'll have one solution.

A. a can be any number