The solution of the given set of simultaneous equations is thus found to be the following set of values:

x = 4,

y = 2,

2 = 7,

w = 1.

From the nature of the derivation of the successive equations it may be seen that solutions have neither been gained nor lost. Hence the solution found is the only solution of the given system of simultaneous equations. Substituting these values for x, y, z, and w, respectively, in the given equations, we obtain the following numerical identities:

1 = 1, (1); 5 = 5, (2); 16 = 16, (3); and 35 = 35, (4).

12. 2x+3y+4≈ +5=0, 2x+3y-4z + 6 = 0, 2x-3y+4z + 7 = 0.

13. 10x8y + z = 40, x + y +22= 5, 3x- 2+5 = 0.

14. 0.3x0.5y = 0.8, 0.4x+0.7% = 1.8, 0.1 y + 0.1 z = 0.3.

15. 0.1x+0.3 y = 1.9, 0.2x+0.4 %= 3.2, 0.5y+0.12

16. x + y = c, y + z = b,

z + x = a.

17.x + y = a + b, y + z = b+c,

z + x = c + a.

3.1.

20. bx + ay + cz = a,

cx + by + az = b,
ax + cy + bz = c.

u + x + y = 6.

26.x + y + z + u = 12,

x + y + z + v = 14,
x + y + u + v = 16,
x + z + u + v = 18,

y + z + u + v = 20.

Systems of Fractional Equations Solved like Equations of the First Degree

46. Certain systems of fractional equations which are linear with reference to the reciprocals of the unknowns may be solved without clearing of fractions before eliminating the unknowns.

Instead of clearing of fractions before eliminating either of the unknowns, in which case we should introduce into the equations terms containing both

x and y, we will eliminate the reciprocals of the unknowns, then from the derived equations obtain values for x and y.

Multiplying members of
equation (2) by 3.

1

1

and and

(1) Unaltered.

By addition,

Hence,

The value of y may be obtained by substituting 5 for x in one of the given equations and solving the resulting equation for y, or by repeating the process of elimination as follows:

Thus, it appears that the following set of values is the solution of the given system of simultaneous equations:

x = 5, y = 3.

Substituting these values in the original equations, we obtain the numerical identities 9 = 9, (1), and 2 = 2, (2).

From the process of derivation it may be seen that solutions have been neither gained nor lost. Hence the set of values found is the only solution of the given system of equations.

47. When solving systems of fractional equations it should be kept in mind that during the process of clearing the fractional

equations of fractions no solutions will be lost, but it may happen. that extra solutions will be introduced. (Compare with Chapter I. § 30, also Chapter XVI. § 4.)

EXERCISE XVII. 6

Solve the following systems of equations which are fractional with. reference to the unknowns, verifying all results obtained :