Elementary Linear Algebra with Applications Howard Anton 9th Ed WILEY

7. Since each of the three given points must satisfy the equation of the curve, we have the
system of equations
ax2

1 + bx1 + c = y1

ax2

2 + bx2 + c = y2

ax2

3 + bx3 + c = y3

If we consider this to be a system of equations in the three unknowns a, b, and c, the
augmented matrix is clearly the one given in the exercise.
9. The solutions of x1 + kx2 = c are x1 = c kt, x2 = t where t is any real number. If these
satisfy x1 + x2 = d, then c kt + t = d, or ( k)t = d c for all real numbers t. In
particular, if t = 0, then d = c, and if t = 1, then = k.
11. If x y = 3, then 2x 2y = 6. Therefore, the equations are consistent if and only if k = 6;
that is, there are no solutions if k 6. If k = 6, then the equations represent the same line,
in which case, there are infinitely many solutions. Since this covers all of the possibilities,
there is never a unique solution.