## Introduction:

We have already discussed about the various methods of solving linear equations in two variables. Let us discuss with graphing systems of linear equations in two variables. We shall also discuss solving systems of linear equations by graphing. By learning the method of linear equations online you will be able to understand the types of solutions.

### How to solve Linear Equations by graphing?

### By solving the linear equations by graphing we can discuss about the type of solution.

### By solving linear equations by graphing we will know if the system of equations has unique solution, no solution or many solution.

### Graphing linear equations in two variables:

#### Algebra of graphing linear Equations:

In algebra of graphing linear equations we find the co-ordinates of the points on each line [ equation ].Since the domain consists of all real numbers, we plug in values for x and find the corresponding values of y.

To graph the linear equation, we find the x and y intercepts by plugging in y = 0 and x = 0 respectively.

Let us discuss the following example.

**Example : Solve the system of linear equations by graphing:**

4x - 3y = 12; x + y = 3

4x - 3y = 12; x + y = 3

#### Solution:

Let us find the x and y intercepts of each equation and record it in the tables given below.4x - 3y = 12

x |
y |

0 | -4 |

3 | 0 |

x+ y = 4

x |
y |

0 | 3 |

3 |
0 |

We are aware that the graph of the linear equations in two variables will be a straight line.

By graphing linear equations the solution to the linear equations will be as follows.

1. If the graph of the pair of linear equations are of intersecting lines then the system of linear equations in two variables is consistent and have unique solution ( one and only one solution)

2. If the graph of the pair of linear equations are of parallel lines then the system of linear equations in two variables is inconsistent and have no solution.

3. If the graph of the pair of linear equations are of coinciding lines then the system of linear equations in two variables is consistent and have infinite number of solutions.

The method of graphing the system of linear equation is also called "Graphing linear equations solver ". This will definitely help us to decide if the system of equations has unique solution, many solutions or no solution.

Let us study the following system of equations and their respective graphs.

#### Example 1: x - 2y = 1 ; x + y = 4

#### Solution:

Let us find the x and y intercepts and record it in the table given below.x - 2y = 1

x |
y |

0 |
1 |

-0.5 | 0 |

x + y = 4

Let us find the x and y intercepts and record it in the tables given below.

x |
y |

0 |
4 |

4 |
0 |

By plotting the points we see that the system of equations

**intersect**at ( 3, 1).

Hence the solution is (3,1)

#### Example 2: 3x - 2y = 6, 6x - 4y = -12

Let us find the x and y intercepts and record it in the tables given below.3x - 2y = 6

x |
y |

0 |
-3 |

2 |
0 |

6x - 4y = -12

x |
y |

0 |
3 |

-2 | 0 |

From the graph we observe that the lines are

**parallel,**Hence we say that the system of equations has "no solution".

#### Example 3: 5x + 2y = 10 ; 10 x + 4 y = 20

Let us find the x and y intercepts and record it in the tables given below.5 x + 2y = 10

x |
y |

0 |
5 |

2 | 0 |

10 x + 4y = 20

x |
y |

-2 | 10 |

4 |
- 5 |

By plotting the above points and graphing we observe that the pair of lines

**coincide**.

This shows that the system of given equation has infinite number of solutions.

Let us discuss one of the above example.In the pair of equations, x - 2y = 1 and x + y = 4,

we got the solution to the pair of equations as (3, 1), as the two lines intersect at this point.

When we graph the linear inequalities, we shade the region satisfying the given inequality.

Let us consider, x - 2y > = 1 and x + y < = 4.

We observe that we get a region between the two common shaded region .This common region is the solution to the inequalities.

Any point in this region will satisfy the given inequality.

Solve the following system of linear equations graphically.

1. x + y = 3 ; 3x + 5y = 15.

2. 3x - y = 2, 9x - 3y = 6

3. 4x - 5y = 4 and 8x - 10 y = 40

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