Question

The pair of equation ax+2y=7 and 3x+by =16 represent parallel lines if​:
a. a=b
b. 3a=2b
c. 2a=3b
d. ab= 6

Answer 1

They can be rewritten as:

ax + 2y - 7 = 0 and

3x + by - 16 = 0

We know that, for the given system of equations to have parallel lines or inconsistent solution,=≠, i.e.,

=

⇒ ab = 6

Answer 2

The pair of equation ax + 2y = 7 and 3x + by = 16 represent parallel lines if ab = 6

Given :

The pair of equation ax + 2y = 7 and 3x + by = 16 represent parallel lines

To find :

The pair of equation ax + 2y = 7 and 3x + by = 16 represent parallel lines if

a. a = b

b. 3a = 2b

c. 2a = 3b

d. ab = 6

Concept :

The pair of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ represent parallel lines if

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

Solution :

Step 1 of 3 :

Write down the given equations

Here the given pair of equations are

ax + 2y = 7 - - - - - - (1)

3x + by = 16 - - - - - - (2)

Step 2 of 3 :

Find the coefficients

ax + 2y = 7 - - - - - - (1)

3x + by = 16 - - - - - - (2)

Comparing the given equations with the equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ we get

a₁ = a , b₁ = 2 , c₁ = 7 and a₂ = 3 , b₂ = b , c₂ = 16

Step 3 of 3 :

Find the required condition

Since the pair of equations represent parallel lines

Thus we get

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

$\displaystyle \sf{ \implies \frac{a}{3} = \frac{2}{b} } \ne \frac{7}{16}$

$\displaystyle \sf{ \implies \frac{a}{3} = \frac{2}{b} }$

$\displaystyle \sf{ \implies ab = 6 }$

Hence the correct option is d. ab = 6

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