Question

if the system of equations 4x+5y=9,8x+ky=18 has infinitely many solutions,then k=​

Answer 1

Answer:

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Answer 2

The value of k = 10

Given :

The system of equations 4x + 5y = 9 , 8x + ky = 18 has infinitely many solutions

To find :

The value of k

Concept :

For the given two linear equations

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

For Infinite number of solutions :

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

Solution :

Step 1 of 3 :

Write down the given system of equations

Here the given system of equations are

4x + 5y = 9 - - - - - (1)

8x + ky = 18 - - - - - (2)

Step 2 of 3 :

Find the coefficients

The equations are

4x + 5y - 9 = 0 - - - - - (1)

8x + ky - 18 = 0 - - - - - (2)

Comparing with the equation

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

a₁ = 4 , b₁ = 5 , c₁ = - 9

a₂ = 8 , b₂ = k , c₂ = - 18

Step 3 of 3 :

Find the value of k

Since there are infinite number of solutions

So we have

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

$\displaystyle \sf{ \implies \frac{4}{8} = \frac{5}{k} = \frac{ - 9}{ - 18} }$

$\displaystyle \sf{ \implies \frac{1}{2} = \frac{5}{k} = \frac{ 1}{ 2} }$

$\displaystyle \sf{ \implies \frac{5}{k} = \frac{ 1}{ 2} }$

$\displaystyle \sf{ \implies k = 10 }$

Hence the required value of k = 10