Question

can someone explain this to me
i need notes and examples about what these are:

one solution

infinite solutions

no solution

Answer 1

Let me start from taking equation in two dimensional plane

A x + B y= C

P x +Q y  = R

For one Solution  

⇒The two lines are intersecting at a single point.

$\frac{A}{P} \neq \frac {B}{Q}\neq \frac{C}{R}$

For example, x + y =4 and x - y=2

For infinite solution

⇒The two lines are Coincident.i.e one lies above the other.

For that,

$\frac{A}{P} = \frac{B}{Q}=\frac{C}{R}$

For example, x + y =3,and 3 x + 3 y =9

For No solution

It means the two lines are parallel i.e they will never intersect.

$\frac{A}{P}=\frac{B}{Q}\neq\frac{C}{R}$

For example, x + y = 11 and 7 x + 7 y = 11

Now consider equation in three dimensional system

Ax +By+Cz=P

Lx +My+Nz=Q

Tx +Uy+Vz=R

You can solve this system of equation by cramers rule or by any other method.

For example taking Cramers rule into consideration

$x =\frac{D_{1}}{D}, y=\frac{D_{2}}{D},  z=\frac{D_{3}}{D}$

For unique Solution or one solution

D ≠ 0

These three lines will intersect at a single point in a three dimensional plane.

x,y,z should have real number as a solution.for example, x=5,y=3,z=4 can be the solution of system of linear equation in three variable satisfying all the equation.

x+y+z=12

2x +y+z =17

x+2y +3z=23

For infinite Solution

These three lines will be Coincident, i.e lies above one another.

I.e,

$D=D_{1}=D_{2} =D_{3}=0$

for example

x +y+z =6

2x +2y +2z=12

3x+ 3y+3z=18 has infinitely many solution.

For no solution

if D=0 and either of $D_{1},$ or $D_{2}$ or$D_{3}$ is non zero ,it has no solution.

The three lines will be parallel to each other.

For example

x + y+z=3

2x +2y+2z =5

4x + 4y + 4z =17

It has no solution.