LINEAR EQUATIONS WITH TWO VARIABLES

THE EQUATIONS WITH TWO VARIABLES x AND y CALLED LINEAR EQUATIONS WITH TWO VARIABLES.

The equation in the form of ax+by+c=0 is called linear equations with two variables.

where a, b and c are constants.

and 

x and y are two variables.

To find two variables i.e. x and y, two equations are required.

Two equations are

ax+by=d

ex+fy=g

Here x and y

and

a,b,c,d,e,f and g are constant.

Solve the following system of linear equations:

{y=2x+4y=3x+2

Now from equation 1 put value of y in equation 2
we get
        y=3x+2
2x+4=3x+2
      4-2=3x-2x
         2=x
Hence x=2
Now putting value of x=2 in equation 1 we get,
y=2x+4
y=2*2+4
y=8

SOLUTION OF THESE TWO EQUATION  x=2 and y=8

 Another example

 x+2y=1      ─⇀ 1

          2x-y = 3        ─⇀ 2

since the coefficient of x and y are both in these two equations

Multiply equation 2 with 2 so that the coefficient of y will be same

i.e.

 x+2y=1            ─⇀ 1

*2(2x-y)=3*2   ─⇀ 2

or

 x+2y=1            ─⇀ 1

4x-2y=6            ─⇀ 2

y=2x+4

y=2x+4

Now adding equation 1 and 2 we get
x+2y+4x-2y=1+6
                5x=7
                 x=7/5
And putting the value of x in equation 1 we get,
      x+2y=1
(7/5)+2y=1
           2y=1-(7/5)
           2y=(5-7)/2
               2y=-2/2
               y=(-1/2)
 Another example

 x+2y=1      ─⇀ 1

          2x-y = 3        ─⇀ 2

since the coefficient of x and y are both in these two equations

Multiply equation 2 with 2 so that the coefficient of y will be same

i.e.

 x+2y=1            ─⇀ 1

*2(2x-y)=3*2   ─⇀ 2

or

 x+2y=1            ─⇀ 1

4x-2y=6            ─⇀ 2

y=2x+4

y=2x+4

Now adding equation 1 and 2 we get
x+2y+4x-2y=1+6
                5x=7
                 x=7/5
And putting the value of x in equation 1 we get,
      x+2y=1
(7/5)+2y=1
           2y=1-(7/5)
           2y=(5-7)/2
               2y=-2/2
               y=(-1/2)

 Another example

 x+2y=1      ─⇀ 1

          2x-y = 3        ─⇀ 2

since the coefficient of x and y are both in these two equations
Multiply equation 2 with 2 so that the coefficient of y will be same
i.e.
 x+2y=1            ─⇀ 1
*2(2x-y)=3*2   ─⇀ 2

or

 x+2y=1            ─⇀ 1

4x-2y=6            ─⇀ 2

y=2x+4

y=2x+4

Now adding equation 1 and 2 we get
x+2y+4x-2y=1+6
                5x=7
                 x=7/5
And putting the value of x in equation 1 we get,
      x+2y=1
(7/5)+2y=1
           2y=1-(7/5)
           2y=(5-7)/2
               2y=-2/2
               y=(-1/2)

What is the value of k for which the system of equations Kx+2y=5 and 3x+y=1 has no solution? Later after solving plz tell me what is no solution?

If two lines are parallel then slope of the line will be same and there will be no solution

kx+2y=5 →1.

3x+y=1  →2.

slope of first line: -k/2

and

slope of second line: -3

For no solution:

-k/2=-3

k=6

No solution means two lines does not intersect each other or two lines are parallel to each other.

METHOD OF SOLVING:

Solve for x and y:

 x+2y=1

3x-2y=7

Solve for x and y:

 x+2y=1

  2x-y=1

For what value of k, system of equations, does x+2y=3&5x+ky=15 have infinite solutions?

given lines x+2y=3 and 5x+ky=15

will have infinite solutions if 

both lines are parallel 

for parallel slope must be equal

-1/2=-5/k

k=10

Work for you:

Find x and y

1.    2x+3y=1

        x+2y=3

2.    x-3y=4

        x+y=2

3.   x+y=2

      x-2y=4