Question

Consider the following pairs of linear equations:
i) 3x+2y=5; 2x+3y=5
ii) 2x-3y=7; 2x-3y=8
Choose the correct option:-
a) the pairs in (i) & (ii) are inconsistent.
b) the pairs in (i) & (ii) are consistent.
c) the pair in (i) is inconsistent & the pair (ii) is consistent.
d) the pair in (i) is consistent & the pair (ii) is inconsistent.

Answer 1

Answer:

Answer (D) is the correct option

Answer 2

Answer:

correct answer is option (d)

the pair in (i) is consistent & the pair (ii) is inconsistent.

Step-by-step explanation:

i) 3x+2y=5; 2x+3y=5

A system of linear equations  $a_1 x +b_1 y +c_1 = 0$ and $a_2x+b_2y +c_2 = 0$ has said to be consistent if it has at least one solution. and inconsistent if it has no solution

The linear equations  $a_1 x +b_1 y +c_1 = 0$ and $a_2x+b_2y +c_2 = 0$has said to be consistent if it has a unique solution or infinite number of solutions.

and inconsistent if it has no solution.

Again we have,

The linear equations  $a_1 x +b_1 y +c_1 = 0$ and $a_2x+b_2y +c_2 = 0$ has

• a unique solution  if  $\frac{a_1}{a_2} \neq \frac{b_1 }{b_2}$
• an infinite number of solutions if    $\frac{a_1}{a_2} = \frac{b_1 }{b_2} = \frac{c_1}{c_2}$
• no solution if if    $\frac{a_1}{a_2} = \frac{b_1 }{b_2} \neq \frac{c_1}{c_2}$

i) 3x+2y=5; 2x+3y=5

Compare the given equations with

$a_1 x +b_1 y +c_1 = 0$ and $a_2x+b_2y +c_2 = 0$

We get,

$a_1 = 3 , b_1 = 2, c_1 =- 5\\a_2 = 2, b_2 = 3, c_2 =- 5$

$\frac{a_1}{a_2} = \frac{3}{2}\\\ \frac{b_1}{b_2} = \frac{2}{3}\\\ \frac{c_1}{c_2} = \frac{-5}{-5}$

we have,  $\frac{a_1}{a_2} \neq \frac{b_1 }{b_2}$,

hence the pair of linear equations 3x+2y=5; 2x+3y=5 has a unique solution. hence the pair of equations is consistent

ii) 2x-3y=7; 2x-3y=8

Compare the given equations with

$a_1 x +b_1 y +c_1 = 0$ and $a_2x+b_2y +c_2 = 0$

We get,

$a_1 = 2 , b_1 = -3, c_1 = -7\\a_2 = 2, b_2 = -3 , c_2 = -8$

$\frac{a_1}{a_2} = \frac{2}{2}\\\ \frac{b_1}{b_2} = \frac{-3}{-3}\\\ \frac{c_1}{c_2} = \frac{-7}{-8}$

we have,  $\frac{a_1}{a_2} = \frac{b_1 }{b_2} \neq \frac{c_1}{c_2}$

hence the pair of linear equations 2x-3y=7; 2x-3y=8 has no solution. hence the pair of equations is inconsistent.

Hence the correct answer is option (d)

the pair in (i) is consistent & the pair (ii) is inconsistent.