System of linear equation is a group of linear equations with a set of variables. A system of linear equation can be in two or three variables, i.e.,

Linear equation with two variables is given by,

Linear equation with three variables is given by,

Here, x, y and z are the three variables.

A solution is a group of variable values that satisfy all set of equations. Solution of a system can have infinite many solution, unique solution or no solution. To solve these system of equation, there are different ways i.e,

Linear equation with two variables is given by,

**Lx + My = N**

Ox + Py = QOx + Py = Q

Linear equation with three variables is given by,

**Lx + My + Nz = O**

Px + Qy + Rz = S

Tx + Uy + Vz = WPx + Qy + Rz = S

Tx + Uy + Vz = W

Here, x, y and z are the three variables.

A solution is a group of variable values that satisfy all set of equations. Solution of a system can have infinite many solution, unique solution or no solution. To solve these system of equation, there are different ways i.e,

**Substitution method****Elimination method****Cramer's rule****Graphing method**

**Step 1:**Observe the set of equation and choose simplest equation to solve for x and y.

**Step 2:**Substitute the simplest equation into other equation and solve for either of the variable.

**Step 3:**After getting the value for one of the variable, find the value of another variable. And check the solution.

Given below are some of the problems based on system of equation.

### Solved Examples

**Question 1:**Solve the following system of equation,

2x + y = 1

-3x - 2y = -3

**Solution:**

Step 1: Given equation :

2x + y = 1

-3x - 2y = -3

As first equation is simplest equation, let us consider.

2x + y = 1

y = 1 - 2x

Step 2: Substitute the first equation into second equation to solve for x.

-3x - 2y = -3

-3x - 2(1 - 2x) = -3

-3x - 2 + 4x = -3

x = -3 + 2

x = -1

Step 3: Let us find the value of y by substituting x value in first equation.

2(-1) + y = 1

-2 + y = 1

y = 1 + 2

y = 3

Therefore, the solution for the given equation is (x, y) = (-1, 3).

**Question 2:**Solve the following system of equation,

-x - 3y = -16

-x + 2y = 9

**Solution:**

Step 1 : Given equation :

-x - 3y = -16

-x + 2y = 9

As second equation is simplest equation, let us consider.

-x + 2y = 9

x = 2y - 9

Step 2 : Substitute the first equation into first equation to solve for y.

-x - 3y = -16

-(2y - 9) - 3y = -16

-2y +9 -3y = -16

-5y = -16 - 9

y = $\frac{25}{5}$

y = 5

Step 3 : Let us find the value of x by substituting y value in second equation.

-x + 2y = 9

-x + 2(5) = 9

x = 10 - 9

x = 1

Therefore, the solution for the given equation is (x, y) = (1, 5).