check whether the pair pf equation 5x-y=7 a d x-y=-1 are consistent or inconsistent?( with step and tell how )
Answers
Hint: For checking whether the pair of linear equations are consistent or inconsistent, we try to obtain values of x and y. If both x and y have a unique value then the system is consistent. The system becomes inconsistent when there exist no values of x and y that satisfy both the equations.
Complete step-by-step answer:
According to the given system of equations, we assign equations corresponding to the expression.
Let the first expression be: 3x+2y=5…(1)
The second expression will be: 2x−3y=7…(2)
Now, we try to eliminate one of the variables x or y by using both the equations.
To do so, we multiply the equation (1) with 3 and multiply the equation (2) with 2.
(3x+2y=5)×39x+6y=15…(3)(2x−3y=7)×24x−6y=14…(4)
Since both the equations have the same value of y, it can be eliminated. Now, adding equation (3) and (4), we get
9x+6y−15+(4x−6y−14)=09x+4x+6y+6y−15−14=013x−29=0x=2913
So, the obtained value of x is 2913.
Putting the value of x in equation 1, we get
3×2913+2y=52y=5−87132y=65−87132y=−2213y=−1113
Hence, the value of y is −1113.
Since there exists a unique value of x and y, therefore the system is consistent.
Note: This problem can alternatively be solved by using the coefficient analysis method for determination of consistent system. In this method the coefficients of x and y i.e. a and b, are compare and if the condition a1a2≠b1b2
is satisfied, then the system is consistent.