# 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 classroom exams are capable of solving system of linear equations of up to three unknowns.

Two equations, two unknowns

This system of equations is in the form

$a_1x + b_1y = c_1$ → equation (1)

$a_2x + b_2y = c_2$ → equation (2)

Three equations, three unknowns

Below is the format of system of equations in three variables.

$a_1x + b_1y + c_1z = d_1$ → equation (1)

$a_2x + b_2y + c_2z = d_2$ → equation (2)

$a_3x + b_3y + c_3z = d_3$ → equation (3)

**Diophantine Equations**

Diophantine equations arise in problems where the number of equations that can be created is less than the number of unknowns, making the system indeterminate. However, unknowns involving this type of system are integers only, and most of the time, excludes zero as a solution. Solving Diophantine system is by trial and error until integers that satisfies all the equations in the system are found.

Example

A man bought 20 pieces of assorted calculators for \$2000. Programmable calculators cost \$300 per unit, the scientific calculators \$150 per unit, and the household type at \$50 per unit. How many household type did he buy?

Solution

*x*= number of programmable calculators

*y*= number of scientific calculators

*z*= number of household type calculators

Total cost is \$2000

$300x + 150y + 50z = 2000$

$6x + 3y + z = 40$ → Equation (1)

Total number of units is 20

$x + y + z = 20$ → Equation (2)

Note:

No other equation can be made from the problem. Although the number of equations is less than the number of unknowns (indeterminate), variables *x*, *y*, and *z* can only hold positive whole numbers, thus, we can solve the problem.

Subtract Equation (2) from Equation (1)

$5x + 2y = 20$

$y = \dfrac{20 - 5x}{2}$

By trial and error:

Try *x* = 1, *y* = 7.5 → not applicable

Try *x* = 2, *y* = 5 → okay!

Try *x* = 3, *y* = 2.5 → not applicable

Try *x* = 4, *y* = 0 → not acceptable

Substitute *x* = 2 and *y* = 5 to Equation (2)

$2 + 5 + z = 20$

$z = 13$ *answer*

Thus, the man bought 2 units of programmable calculators, 5 units of scientific calculators, and 13 units of household type calculators.

**Other forms of system of equations**

There are many types of system of equations. They may contain quadratic equations, it may be in exponential form, or may contain logarithm, and so on. The solution, however, can be unified into one, that is, by solving the equations in the system simultaneously.

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