A system of equations is a set of equations that we deal with all together at once. Linear equations are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables. If these two linear equations intersect, that point of intersection is called the solution to the system of linear equations.

The solution of system of linear equations in two variables is the point of intersection (x, y) of the graphs of the line.

**The algebraic methods** of solving a system of linear equations, the substitution and elimination methods can lead to an exact solution of the system. This against the graphical method of solving the system often gives approximate solutions as it is difficult to read the coordinates accurately. But still, the **graphical method provides a visual proof to the existence of the solution**.

The solution of system of linear equations in two variables is the point of intersection (x, y) of the graphs of the line.

The system has exactly one solution

Systems have 1 and only 1 solution when the two lines have different slope. If the two lines have different slopes then eventually at some point they must meet or

System has no solutions

The Systems have no solution when the

The system has infinite solutions

The Systems have infinite solutions when the

Step 4:

$y + x = 3$ and $3y - x = 5$

Write both the equations in slope intercept form isolating y on one side of the equation

$y = - x + 3$ and $y$ = $\frac{x + 5}{3}$

$y = - x + 3$

x |
y |

1 | 2 |

0 | 3 |

-1 | 4 |

$y$ = $\frac{x + 5}{3}$

x |
y |

1 |
2 |

-2 | 1 |

4 | 3 |

Even though two points are sufficient to draw a line, three points are taken to ensure there is no error done in graphing.

The graph of the lines are shown with the points marked. The green line represents the equation y = - x + 3 and the purple line is the graph of 3y - x = 5.

=> Lines intersect at the point (1, 2)

Hence the solution of the system is (1, 2) where x =1 and y = 2.

$y - x = 3$ and $y + x = -1$

Find the x and y intercepts of the given lines

y = 3 + x and y = -1 - x

Make a table of values to be plotted for both the equations

y = 3 + x

x | y |

0 |
3 |

-3 |
0 |

-2 |
1 |

and y = -1 - x

x |
y |

0 |
-1 |

-1 |
0 |

-2 |
1 |

The intercepts are plotted and the graph of the lines made joining the respective intercepts.The green line is the graph of y - x = 3 and the the purple line represents y + x = -1.

The above graph shows the point of intersection as (-2,1)

Hence the solution to the system is