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System of Linear Equations

A system of linear equations is the set of linear equations that we actually deal together to reach the solution.

For example: y = 2x + 3 and y = 3x + 2, when we simplify these two equations, we see that (1, 5) lies on both the equations.
The values of variables "x" and "y" which satisfy the equation ax + by + c = 0, is termed as its solution.
Note that every linear equation in two variables has unlimited number of solutions.

A linear equation is a one-degree algebraic equation which gives a straight line when we plot them on a graph. It is usually of the form ax + b = 0, where a, b are real numbers and "a" is not equal to zero. This is called a linear equation in one variable x.

If we have two variables, x and y, then this equation gets into the form of ax + by + c = 0, where 'a' and 'b' are not equal to zero.
This equation is termed as a linear equation in two variables, x and y.

To solve the above equation, the first method is the Graphical Method, in which we construct a graph of linear equation ax + by + c = 0, which is an equation of degree one.
Thus, its graph would be a straight line and every point lying on the line will be its solution.

The other method for solving the set of simultaneous linear equations, where a pair of values of x and y which satisfies each of the given system of these equations and is called a solution of the system.

A system of two simultaneous equations would be considered as consistent, if it is having at least one solution, and if there is no existing solution, then it would be construed as inconsistent.
We can solve a set of simultaneous equations by graphical method, in which, three cases arises :
(a) When the two lines are parallel, the system has no solution, hence the system is inconsistent.
(b) When the two lines are coincident, the system has an infinite number of solutions, hence the system is consistent.
(c) When the two lines are intersecting, the system has a unique solution, hence the system is consistent.
We can solve the simultaneous equations by algebraic methods as well, which includes
(a) Substitution method (substituting one variable’s value into other equation.)
(b) Elimination Method (eliminating one variable)
(c) Cross Multiplication Method.

Related Calculators
Linear System of Equations Solver

Solving System of Linear Equations

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Solved Examples

Question 1: Solve x + 2y = 5 and 2x – 3y = 3.
Let us solve this system of linear equations by substitution method.
From first equation, we can write
x = 5 – 2y
Plugging this value in second equation, we get
2 (5 – 2y) – 3y = 3
We can write this as 10 – 4y – 3y = 3
This gives y = 1
Substituting the value of y in x + 2y = 5, we get
x + 2 = 5
x = 3
Therefore, the solution is x = 3, y = 1.


Question 2: Solve x + y = 1 and x – y = 3.
Let us solve this system by elimination method.
First add these two equations
x + y + x – y = 1 + 3
which leads to the elimination of y
x = 2.
Plug this value in the equation x + y = 1
2 + y = 1
y = 1 – 2 = - 1 
Therefore, the solution is x = 2 and y = -1.

Question 3: Solve a + b = 4 and 2a – b = 5.
First we can add these two equations.
a + b + 2a – b = 4 + 5.
This gives a = 3
We plug this value in the equation a + b = 4. Then 3 + b = 4
b = 1
Therefore, solution is a = 3 and b = 1.

More topics in  System of Linear Equations
Solving Systems of Linear Equations
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