5. If an equation were proposed involving three unknown quantities, as ax + by + cz = d, by transposition we have ax + by = d - cz, and, putting d - cz = c', ax + by = c'. From this last equation we may find values of x and y of this form, x = mr + nc', y = mr + n'c', or x = mr + n(d - cz), y = m'r + n'(d - cz); where z and r may be taken at pleasure, except in so far as the values of x, y, z may be required to be all positive; for from such restriction the values of z and r may be confined within certain limits to be determined from the given equation. For more advanced treatment of linear indeterminate equations see COMBINATORIAL ANALYSIS. 6. We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y^2 = a + bx + cx^2, where x is to be found, so that y may be a rational quantity. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature. _Case_ 1. Let a be a square number; then, putting g^2 for a, we have y^2 = g^2 + bx + cx^2. Suppose [root](g^2 + bx + cx^2) = g + mx; then g^2 + bx + cx^2 = g^2 + 2gmx + m^2 x^2, or bx + cx^2 = 2gmx + m^2 x^2, that is, b + cx = 2gm + m^2x; hence 2gm - b cg - bm + gm^2 x = --------, y = [root](g^2 + bx + cx^2) = --------------, c - m^2 c - m^2 _Case_ 2. Let c be a square number = g^2; then, putting [root](a + bx + g^2 x^2) = m + gx, we find a + bx + g^2x^2 = m^2 + 2mgx + g^2 x^2, or a + bx = m^2 + 2mgx; hence we find m^2 - a bm - gm^2 - ag x = -------, y = [root](a + bx + g^2 x^2) = --------------. b - 2mg b - 2mg _Case_ 3. When neither a nor c is a square number, yet if the expression a + bx + cx^2 can be resolved into two simple factors, as f + gx and h + kx, the irrationality may be taken away as follows:-- Assume [root](a + bx + cx^2)=[root]{(f + gx)(h + kx)} = m(f + gx), then (f + gx)(h + kx) = m^2(f + gx)^2, or h + kx = m^2(f + gx); hence we find fm^2 - h (fk - gh)m x = --------, y = [root]{(f + gx)(h + kx)} = ----------; k - gm^2 k - gm^2 and in all these formulae m may be taken at pleasure.