1. How to express your linear equations in the matrix form AX = B.
2. You know how to find the determinant of A when A is a 2x2 matrix.
3. You understand that the equations have no unique solution when the determinant of A, det(A), is equal to zero.
In your problem, det(A) = 2 - ab.
det(A) = 0 => 2 - ab = 0 => ab = 2.
So there will be NO unique solution when ab = 2. Therefore there WILL be a unique solution when .
When there's no unique solution, that means that either there is an infinite number of solutions or no solution at all. To test which of these happens when ab = 2, you might do something like the following:
. Substitute this into you equations and do some re-arranging:
Equation (1): .
Equation (2): .
Now compare equations (1) and (2). You can see that when the equations are the same. One equation and two unknowns => infinite number of solutions.
So when ab = 2 AND a = -5, there an infinite number of solutions.
It follows that there will be no solution at all when ab = 2 AND .
When a = -5 and ab = 2, . So the equations boil down to -5x + 2y = 5. The form of the infinite number of solutions can be expressed in several ways. To see where the books answer came from, note that if you let y = t, where t can be any real number:
As to where the unique solution comes from when :
x + by = -1 .... (1)
ax + 2y = 5 .... (2)
Solve using the elimination method:
Multiply equation (1) by a, subtract it from equation (2) and then solve for y:
(2 - ab)y = 5 + a => ......
Then sub y into either equation and solve for x.