Advertisements
Advertisements
Question
Discuss the nature of solutions of the following system of equations
`(y + z)/4 = (z + x)/3 = (x + y)/2`; x + y + z = 27
Solution
`(y + z)/4 = (z + x)/3`
3y + 3z = 4z + 4x
– 4x + 3y + 3z – 4z = 0
– 4x + 3y – z = 0
4x – 3y + z = 0 ...(1)
`(z + x)/3 = (x + y)/2`
2z + 2x = 3x + 3y
– 3x + 2x – 3y + 2z = 0
– x – 3y + 2z = 0
x + 3y – 2z = 0 ...(2)
x + y + z = 27 ....(3)
4x – 3y + z = 0 ....(1)
x + 3y – 2z = 0 ....(2)
(1) + 2 ⇒ 5x + 0 – z = 0 ....(4)
(3) × 3 ⇒ 3x + 3y + 3z = 81 ....(3)
(2) × 1 ⇒ x + 3y – 2z = 0 ....(2)
(–) (–) (+)
(3) – 2 ⇒ 2x + 5z = 81 ....(5)
(4) × (5) ⇒ 25x – 5z = 0 ....(4)
(5) × 1 ⇒ 2x + 5z = 81 ....(5)
(4) + 5 ⇒ 27x = 81
x = `81/27` = 3
Substituting the value of x in (5)
6 + 5z = 81
5z = 81 – 6
5z = 75
z = `75/5` = 15
Substituting the value of x = 3
and z = 15 in (3)
3 + y + 15 = 27
y + 18 = 27
y = 27 – 18
= 9
The value of x = 3, y = 9 and z = 15
This system of equations have unique solution.
APPEARS IN
RELATED QUESTIONS
Solve the following system of linear equations in three variables
x + y + z = 5; 2x – y + z = 9; x – 2y + 3z = 16
Solve the following system of linear equations in three variables
`1/x - 2/y + 4 = 0; 1/y - 1/z + 1 = 0; 2/z + 3/x = 14`
Solve the following system of linear equations in three variables
x + 20 = `(3y)/2 + 10` = 2z + 5 = 110 – (y + z)
Discuss the nature of solution of the following system of equation.
x + 2y – z = 6; –3x – 2y + 5z = –12; x – 2z = 3
Discuss the nature of solutions of the following system of equations
2y + z = 3(– x + 1); – x + 3y – z = – 4; 3x + 2y + z = `-1/2`
Vani, her father and her grandfather have an average age of 53. One-half of her grandfather’s age plus one-third of her father’s age plus one-fourth of Vani’s age is 65. Four years ago if Vani’s grandfather was four times as old as Vani then how old are they all now?
The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three-digit number?
There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹ 105. When first 2 sorts are interchanged in their numbers its value will be increased by ₹ 20. Find the number of currencies in each sort
A system of three linear equation in three variables is inconsistent if their planes
