Consistent and Dependent Systems

x - 2y = 4

x + 3y = -2

This system is: a. consistent independent

b. consistent dependent

c. inconsistent

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`x - 2y = 4` ...............(i)

`x + 3y = -2` ...............(ii)

From (i) we get:

`x=2y+4` .............(iii)

Substituting the value of x in equation (ii) we get:

`2y+4+3y=-2`

`rArr 5y=-6`

`rArr y=-6/5`

Putting the value of y in (iii) we get:

`x=2*-6/5+4=-12/5+4=8/5`

Hence, the solution is `(8/5,-6/5)` or `(1.6,-1.2)` .

Since, the system of equations has a single solution or one ordered pair that is a solution to both equations, it is a consistent independent system.

The graphical solution is the intersecting point of the two straight lines.

The red line represents equation (i) and the blue line equation (ii), both intersecting at a single point (1.6,-1.2).

Therefore, the correct answer is **option a. consistent independent**.

**Sources:**

`a_1x+b_1y=c_1`

`a_2x+b_2y=c_2`

This syetem of equations is consistent if

`a_1/a_2!=b_1/b_2`

Thus in the given system of equations,

`x-2y=4`

`x+3y=-2`

`1/1!=(-2)/3`

`=>` System of equations is consistent.

Option A is corrct choice.

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