In linear algebra, linear equations is one of the most important topics. We deal with various linear equations with number of variables such as linear equations with one variable, linear equations with two variable, linear equations with three variable etc. Solving linear equations means to find the solutions for a particular given set of linear system which must satisfy all of the equations. Solution for linear equation in one variable is a single point, or a line or the empty set. While dealing with set of two linear equations, we have different sets of solutions like: unique solution, no solution or infinite many solutions.

What is the solution of a system of linear equations? A system of linear equations is a set of two or more linear equations. The pair of 2 linear equations involves two variables with degree one. Each linear equation is a straight line on the coordinate plane (xy-plane). Any set of solutions which satisfy the given equations is the solution set for the given equations.

Algebraic and Graphical, two ways to get the solution set for the system of linear equations.

Graphing method, Substitution Method, Matrix method and Elimination method are the most important algebraic methods.

Graphing method, Substitution Method, Matrix method and Elimination method are the most important algebraic methods.

The lines are parallel (No Solution)

Lines intersect at one point. (Unique Solution)

All equations represent same line ( Infinitely many Solutions)

Lines intersect at one point. (Unique Solution)

All equations represent same line ( Infinitely many Solutions)

7x - 8y = 12

7x = 2 (6 + 4y)

7x - 8y = 12

7x = 2 (6 + 4y)

7x = 2 (6 + 4y)

7x = 12 + 8y

7x = 12 + 8y

7x - 8y = 12

7x - 8y = 12 ...(1)

7x - 8y = 12 ...(2)

-1(7x - 8y = 12)

-7x + 8y = -12

7x - 8y = 12

-7x + 8y = -12

__________________

0x + 0y = 0

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That is 0 = 0

Question 3

(a) 4x - 3y = 12; x + y = 3

(b) y + 4 = -2x and y = -2x - 4

(a) 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 |

The coordinates of intercepts are ( 0, -4) and ( 3, 0)

x+ y = 3

x | y |

0 | 3 |

3 | 0 |

The coordinates of intercepts are (0,3) and ( 3, 0)

**Graphical representation: **

From the graph we observe that the two lines intersect at ( 3,0) which is the solution to the above system of equations.

(b) y + 10 = -5x and y = -5x - 10

y + 10 = -5x

x | y |

0 | -10 |

-2 | 0 |

The coordinates of intercepts are (0, -10) and (-2, 0)

y = -5x - 10

The coordinates of intercepts are (0, -10) and (-2, 0)

Graphical representation:

From the graph we observe that, both the lines coincide. So system have infinite many solutions.

y = -5x - 10

x | y |

0 | -10 |

-2 | 0 |

Graphical representation:

From the graph we observe that, both the lines coincide. So system have infinite many solutions.