Explanation:

Simultaneous linear equations are equations with two or more variables that are to be solved at the same time. These equations can be represented by lines, and the solutions represent the points where these lines intersect.

When two lines intersect at one point, there is only one solution to the system of equations. When two lines are parallel, there is no solution to the system of equations. However, when two lines are coincident, they overlap each other and have infinite solutions.

Example:

Consider the system of equations:

2x + 3y = 6
4x + 6y = 12

We can solve this system of equations by using elimination or substitution method.

Using the elimination method, we can multiply the first equation by 2 and subtract the second equation from it to eliminate x, which gives:

4x + 6y = 12
- (4x + 6y = 12)
-----------------
0x + 0y = 0

This equation is always true, which means that the two equations are equivalent. Therefore, they represent the same line, and there are infinitely many solutions to this system of equations.

Using the substitution method, we can solve for y in the first equation and substitute it into the second equation, which gives:

y = (6 - 2x)/3
4x + 6((6 - 2x)/3) = 12

Simplifying the second equation, we get:

4x + 4x = 12

Which gives:

x = 3/2

Substituting this value of x into the first equation, we get:

2(3/2) + 3y = 6

Simplifying, we get:

3y = 3

Which gives:

y = 1

Therefore, the solution to this system of equations is (3/2, 1). However, this is just one solution, and there are infinitely many solutions to this system of equations since the two lines are coincident.

When the two lines are the same line, then the system should have infinite solutions. It means that if the system of equations has an infinite number of solution, then the system is said to be consistent.