✨ Complete 3-Variable Equation Solver ✨

Problem 1(i): Solve the system

x + y + z = 5

2x − y + z = 9

x − 2y + 3z = 16

Problem 1(ii): Solve the system

(1/2)(x - y) + (1/4)(y + z) = 0

(1/1)(-x + z) + (1/0)(z + x) → Undefined

(2/3)(x + y) = 14

Note: The second equation contains division by zero (1/0), which makes the system undefined. Please check the original equations for possible typos.

Problem 1(iii): Solve the system

x + 20 = (3/2)y + 10

(3/2)y + 10 = 2z + 5

2z + 5 = 110 – (y + z)

Problem 2(i): Nature of Solutions

x + 2y − z = 6

−3x − 2y + 5z = −12

x − 2z = 3

Problem 2(ii): Nature of Solutions

2y + z = 3(−x + 1)

−x + 3y − z = −4

3x + 2y + z = −1/2

Problem 2(iii): Nature of Solutions

(y + z)/4 = (z + x)/3 = (x + y)/2

x + y + z = 27

Problem 3: Age Word Problem

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, Vani's grandfather was four times as old as Vani. Find their current ages.

Problem 4: Digit Problem

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 original number. If the hundreds digit plus twice the tens digit is equal to the units digit, find the original number.

Problem 5: Currency Problem

There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹105. When the first two types are interchanged in their numbers, the value increases by ₹20. Find the number of each currency type.

Thinking Corner 🤔

1. The number of possible solutions when solving system of linear equations in three variables are _____.

2. If three planes are parallel then the number of possible point(s) of intersection is/are _____.