Question

how many solution does the pair of equations 3x + 4y + 2 = 0 , 4x = 5y - 13 have ? give reason

Answer 1

The pair of equations 3x + 4y + 2 = 0 , 4x = 5y - 13 have unique solution

Given :

The pair of equations 3x + 4y + 2 = 0 , 4x = 5y - 13

To find :

The number of solutions of the pair of equations

Concept :

For the given two linear equations

$\displaystyle \sf{ a_1x+b_1y+c_1=0 \: and \: \: a_2x+b_2y+c_2=0}$

Consistent :

One of the Below two condition is satisfied

1. Unique solution :

$\displaystyle \sf{ \: \frac{a_1}{a_2} \ne \frac{b_1}{b_2} }$

2. Infinite number of solutions :

$\displaystyle \sf{ \: \frac{a_1}{a_2} = \frac{b_1}{b_2} = \: \frac{c_1}{c_2}}$

Inconsistent :

No solution

$\displaystyle \sf{ \: \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \: \frac{c_1}{c_2}}$

Solution :

Step 1 of 2 :

Write down the given pair of equations

Here the given pair of linear equations are

3x + 4y + 2 = 0 - - - - - (1)

4x = 5y - 13

⇒ 4x - 5y + 13 = 0 - - - - - (2)

Step 2 of 2 :

Find the number of solutions

3x + 4y + 2 = 0 - - - - - (1)

4x - 5y + 13 = 0 - - - - - (2)

Comparing with the equations

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 we get

a₁ = 3 , b₁ = 4 , c₁ = 2 and a₂ = 4 , b₂ = - 5 , c₂ = 13

Now we have ,

$\displaystyle \sf\frac{a_1}{a_2} = \frac{3}{4}$

$\displaystyle \sf \frac{b_1}{b_2} = \frac{4}{ - 5} = - \frac{4}{5}$

$\displaystyle \sf \frac{c_1}{c_2} = \frac{2}{13}$

Thus we get ,

$\displaystyle \sf\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \ne \: \frac{c_1}{c_2}$

Hence the pair of equations 3x + 4y + 2 = 0 , 4x = 5y - 13 have unique solution

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Answer 2

Answer:

The given pair of equations has a Unique solution and the solution is Consistent.

Step-by-step explanation:

Given pair of equations are

 3x+4y+2 = 0 and 4x = 5y-13

can be written as

    3x+4y+2 = 0   -----------(i)

    4x-5y+13 = 0  ------------(ii)

 to find the pair of equations have no. of solution we have conditions such as

Let ax+by+c = 0 and px+qy+r = 0 be the two pair of equations

 then these equations may have consistent solution or inconsistent solution.

Consistent solution:

For a solution to be consistent the pair of equations may have a unique solution or an Infinite no. of solutions.

• Unique solution:

        If the condition a/p ≠ b/q then the pair of equations has unique solution.

• Infinite solution:

      If a/p = b/q = c/r then the pair of equations has Infinite no. of solutions.

Inconsistent solution:

For a solution to be Inconsistent it should satisfy the condition given below.

• No solution:

         If  a/p = b/q ≠ c/r then the pair of equations has no solution and it is Inconsistent.

 Now compare

    3x+4y+2 = 0 and 4x-5y+13 = 0  with

     ax+by+c = 0 and px+qy+r = 0

        Here a=3, b=4, c=2, p=4, q=-5, r=13

         Now we check the conditions

        a/p = 3/4, b/q = 4/-5, c/r = 2/13

             Since, a/p ≠ b/q

Hence the given pair of equations has Unique solution.

The solution is Consistent.

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