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 linear 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 linear 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

Other forms of system of equations

There are many types of system of equations. It may contain quadratic equations, or 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.