7.

The curve y=ax2+bx+c passes through the points (x1,y1), (x2, y2), (x3,y3). Show that the coefficients a, b, c are a solution of the system of equations with augmented matrix
[ x12 x1 1 y1 ]
[ x22 x2 1 y2 ]
[ x32 x3 1 y3 ]

Solution.

Since the points are on the curve, their coordinates satisfy the equation of the curve. Thus we have: 

 	               x12 a+x1 b + c = y1
	               x22 a+x2 b + c = y2
	               x32 a+x3 b + c = y3

This is a system of linear equations with unknowns a, b, c and augmented matrix as in the formulation of the problem.

8.

For which values of the constant k does the following system of equations have no solutions? Exactly one solution? Infinitely many solutions?
			                     x -  y = 3
			                    2x - 2y = k

Solution.

Multiplying the first equation by 2 we get 2x-2y=6. Comparing with the second equation we conclude that if k is not 6, the system has no solutions. Suppose k=6. Then replacing the second equation by the difference of the the second equation and twice the first equation, we get the following equivalent system:

			                  x -  y = 3
			                 0x - 0y = 0

This system of equations has infinitely many solutions given by the formula: 

			                  x = 3+t
			                  y = t

where t is an arbitrary parameter. Thus if k=6, the system has infinitely many solutions.

Answer: the system has no solutions if and only if k is not equal to 6. It has infinitely many solutions if and only if k=6. And it never has exactly one solution.

Alternative solution: consider these equations as equations of lines on the plane. The lines are parallel (no solutions) iff k is not 6. If k=6 the lines coincide, so the intersection is a line (infinitely many solutions)