System of linear equations is a set or collection of variables involving the same set of variables. A linear equation in variables $x_{1},x_{2},........,x_{n}$ is an equation of the form $a_{1}x_{1},a_{2}x_{2},........,a_{n}x_{n}$ = b. It will have two or more equations in one or more variables. System of linear equations are used to make accurate predictions in real world problems.

System of Linear Equations Definition

A system of linear equations is one which may be written in the form:

$a_{11}x_{1}$ + $a_{12}x_{2}$ + .........+ $a_{1n}x_{n}$ = $b_{1}$
$a_{21}x_{1}$ + $a_{22}x_{2}$ + .........+ $a_{2n}x_{n}$ = $b_{2}$
...... ........
....... .......
$a_{m1}x_{1}$ + $a_{m2}x_{2}$ + .........+ $a_{mn}x_{n}$ = $b_{m}$

All of the coefficients $a_{ij}$ and $b_{i}$ are assumed to be known constants. All the $x_{i}$'s are assumed to be unknowns that we are to solve for.

A system of linear equations means two or more linear equations. If two linear equations intersect then the point of intersection is called the solution to the system of linear equations.

Consistent and Inconsistent System of Linear Equations

Solution of the system of equations is an ordered pair that satisfies each equation in the system. If there are three equations with three variables then the ordered pair is (a, b, c). System of linear equations are of three types with
• One Solution
• Infinite Solution and
• No Solution
Consistent System

If the system has at least one solution then it is known as consistent system of linear equations and they Intersect at one point and have one solution. Also known as independent system of equations.

Inconsistent System

If the system has no solution then it is said to be inconsistent system and has no solutions. In this case the lines will be parallel and never intersect.

System of Linear Equations in Two Variables and Three Variables

A linear  equations with two variables

ax + by = p
cx + dy = q

where any of the constants can be zero with the exception that each equation must have at least one variable in it. The variables should be to the first power only.

Similarly an equation of three variables with three equations can be written as

ax + by = p
cx + dy = q
ex + fy = r
Here also the variables should be in first power only.

System of Linear Equations Word Problems

Solved Example

Question: A total of 12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is 1,180. How much of the 12,000 is invested at each rate?
Solution:

Let x: Investment in fund for 9 years.
Let y: Investment in fund for 11 years.
Then the above statement can be written in the mathematical form as:
x  +      y    =     12000
0.09x + 0.11y     =      1180

Multiply the first equation by -0.09, we get
-0.09x   -   0.09y  = -1080
0.09x    +  0.11y  =  1180
_________________________
0.02y =  100
______________________
$\Rightarrow$  y = 5000

To find x, substitute y =  5000 in   x + y = 12000
x + 5000 =    12000
$\Rightarrow$  x = 12000 - 5000
$\Rightarrow$  x = 7000

Therefore, x = 7000 is the investment for 9 years and y = 5000 is the  investment for 11 years.