The sum of three numbers is 6. If we multiply the third number by 2 and add the first number to the result we get 7. By adding second and third numbers to three times the first number we get 12. Find the numbers using rank method

A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below

Ounces per pound of Nutrient
Food	P	Q	R
A	1	2	5
B	3	1	1
C	4	2	1

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R? (Cramer's rule).

For what values of k, the system of equations kx+ y+z = 1, x+ ky+z= 1, x+ y+kz = 1 have
(I) Unique solution
(ii) More than one solution
(iii) no solution

Using determinants, find the quadratic defined by f(x) = ax² + bx + c if
f(1) = 0,
f(2) = - 2 and
f(3) = -6.

A new transit system has just gone into operation in a city. Of those who use the transit system this year, 10% will switch over to using their own car next year and 90% will continue to use the transit system. Of those who use their cars this year, 80% will continue to use their cars next year and 20% will switch over to the transit system. Suppose the population of the city remains constant and that 50% of the commuters use the transit system and 50% of the commuters use their own car this year,
(i) What percent of commuters will be using the transit system after one year?
(ii) What percent of commuters will be using the transit system in the long run?

Show that the equations are consistent and have a unique. solution. Solve, them using rank method 2x-y + z = 7, 3x + y- 5z = 13, x + y + z = 0.

Find k if the equation x + 2y 3 = -2, 3x - y - 2z = 1 and 2x + 3y 5z = k are consistent

Find k if the equations x + y + z = 3, x + 3y + 2z = 6, x + 5y + 3z = k are in consistènt.

Verify whether the given system of equations is consistent. If it is consistent solve, them 2x + 5y + 7z = 52, x + y + z = 9, 2x + y - z = 0.

Show that the equations x + y + z = 6, x + 2y + 3z 14, z + 4y + 72 = 30 are consistent and solve them.

Investigate for what values of \(\lambda\) the simultaneous equations x+y+z = 6, x +2y + 3z = 10, x + 2y + \(\lambda\)z have
(i) no solution
(ii) unique solution
(iii) infinite number of solutions

Discuss the solution of the system of equation for all values of \(\lambda\) \(x+y+z=2,2 x+y-2 z=2, \lambda x+y+4 z=2\)

For what values of k, the system of equations kx + y.+ z = 1, x+ ky + z = 1, x + y + kz = 1, have
(i) unique solution
(ii) more than one solution
(iii) No solution

Examine the consistency of the system of equations. If it is consistent then solve the same.
\((i) 4 x+3 y+6 z=25, x+5 y+7 z=13, 2 x+9 y+z=1\)
\((ii) x-3 y-8 z=-10,3 x+y-4 z=0, 2 x+5 y+6 z-13=0\)

Solve the equations by determinant method. 
(i) x + 2y + 5z = 23, 3x + y + 4z = 26, 6x + y + 7z = 47
(ii) 2x + 2y - z - 1 = 0, x + y -7 = 0, 3x + 2y - 3z = 1
(iii) \(\frac{1}{x}+\frac{2}{y}-\frac{1}{z}=1, \frac{2}{x}+\frac{4}{y}+\frac{1}{z}=5 ; \frac{3}{x}-\frac{2}{y}-\frac{2}{z}=0\)