Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

DETERMINANTS

383

the first of the equations (1). In like manner it can be shown
that they satisfy the other two equations.
This completes the proof that equations (1), provided I a b c
= 0, have a unique solution, which is given by Cramer's rule.
Both the proof and the rule can be generalized to the case of
any number of simultaneous linear equations in the same number of unknowns. We state the result in general form.
THEOREM 10. A number of simultaneous linear equations in
the same number of unknowns, for which the determinant of the
coefficients of the unknowns does not vanish, has one and only one
solution, which is given by Cramer's rule.*
EXERCISES
1. Deduce the value of y given by (2).
2. Prove that the values of x, y, z given by (2) actually
satisfy the third of equations (1).
3. Give Cramer's rule for four simultaneous linear equations
in four unknowns. First write down the equations and then
the formulas analogous to formulas (2). No proof is required.
Solve the following systems of simultaneous equations.
2x- y+3z+ t= 6, 3y-4z-+2t- 4=0,
— x+2y+4z+3t=-6, 2x +3z-4t+ 3=0,
3x-2y — z+4t-=-1, - -4x+2y +3t+11 = 0,
4x -3y-Sz-4t = 8. 3x-4y-2z - 5=0.
9. Three Equations inTwoUnknowns. Compatibility. The
three linear equations,
alx + bly + cl = 0,
(1) a2x + by - c = 0,
a3x + by + c = 0,
* In case the determinant of the coefficients does vanish, the facts are
more complex. For two equations in two unknowns, they are given in
the footnote on p. 360. For a treatment of the general case, cf. BOcher,
Introduction to Higher Algebra, Ch. IV.