# Simultaneous Equations

## Number of Civil, Electrical, and Mechanical Engineers 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.

## Example 07 - Simultaneous Non-Linear Equations of Two Unknowns

**Problem**

Solve for $x$ and $y$ from the given system of equations.

$\dfrac{3}{x^2} - \dfrac{4}{y^2} = 2$ ← Equation (1)

$\dfrac{5}{x^2} - \dfrac{3}{y^2} = \dfrac{17}{4}$ ← Equation (2)

## Example 06 - Simultaneous Non-Linear Equations of Two Unknowns

**Problem**

Solve for $x$ and $y$ from the given system of equations.

$x^2y + y = 17$ ← Equation (1)

$x^4y^2 + y^2 = 257$ ← Equation (2)

## Smallest number for given remainders

## Example 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)

## Example 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)

## Example 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)

## 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.

## Example 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)