## 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)

**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)

**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.

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