## 10 - Smallest number for given remainders

Problem
Find the smallest number which when divided by 2 the remainder is 1, when divided by 3 the remainder is 2, when divided by 4 the remainder is 3, when divided by 5 the remainder is 4, and when divided by 6 the remainder is 5.

## 09 - Number of CEs, EEs, and MEs and their Average Ages

Problem
In an organization there are CE’s, EE’s and ME’s. The sum of their ages is 2160; the average age is 36; the average age of CE’s and EE’s is 39; the average age of EE’s and ME’s is 32 and 8/11; the average age of the CE’s and ME’s is 36 and 2/3. If each CE had been 1 year older, each EE 6 years and each ME 7 years older, their average age would have been greater by 5 years. Find the number of CE, EE, and ME in the group and their average ages.

## 05 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following simultaneous equations.

$x^2 - yz = 3$   ←   Equation (1)

$y^2 - xz = 4$   ←   Equation (2)

$z^2 - xy = 5$   ←   Equation (3)

## 04 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following system of equations.
$x(y + z) = 12$   →   Equation (1)

$y(x + z) = 6$   →   Equation (2)

$z(x + y) = 10$   →   Equation (3)

## 03 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Find the value of x, y, and z from the given system of equations.
$x(x + y + z) = -36$   →   Equation (1)

$y(x + y + z) = 27$   →   Equation (2)

$z(x + y + z) = 90$   →   Equation (3)

## 02 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Find the value of x, y, and z from the following equations.
$xy = -3$   →   Equation (1)

$yz = 12$   →   Equation (2)

$xz = -4$   →   Equation (3)

## 01 - Simultaneous Non-Linear Equations of Three Unknowns

Problem
Solve for x, y, and z from the following simultaneous equations.

$z^x \, z^y = 100\,000$   ←   equation (1)

$(z^x)^y = 100\,000$   ←   equation (2)

$\dfrac{z^x}{z^y} = 10$   ←   equation (3)

## System of Equations

System of Linear Equations
The number of equations should be at least the number of unknowns in order to solve the variables. System of linear equations can be solved by several methods, the most common are the following,

1. Method of substitution
2. Elimination method
3. Cramer's rule

Many of the scientific calculators allowed in board examinations and class room exams are capable of solving system of linear equations of up to three unknowns.