Question

8. Which of the following pair of linear equations represent parallel lines ?

(A)2x + 3y + 5 = 0
3x + 2y – 5 = 0

(B) x-y-10= 0
2x –2y + 15 = 0

(C) 2x + 5y +2=0
4x - 10y+4=0

(D) x–y=10
x + y = 14

if you don't know please don't answer, if anyone gives silly or irrevelent answers I will report them.
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Answer 1

Answer:

answer is B

Step-by-step explanation:

x-y-10=0 has slope =1

similarly 2x-2y +15 =0 has slope of -1

so they have same slope this implies they have same inclination from axis which indicates that they are parallel lines.

Answer 2

Answer:

The pair of linear equations x - y - 10 = 0 and 2x - 2y + 15 = 0 represents parallel lines.

Option B)

Step-by-step-explanation:

We have given the pairs of linear equations.

We have to find the pair of linear equation which represents parallel lines.

A)

The given linear equations are

2x + 3y + 5 = 0 and 3x + 2y - 5 = 0

Now,

2x + 3y + 5 = 0 - - ( 1 )

Comparing with ax + by + c = 0, we get,

• a₁ = 2
• b₁ = 3
• c₁ = 5

Now,

3x + 2y - 5 = 0 - - ( 2 )

Comparing with ax + by + c = 0, we get,

• a₂ = 3
• b₂ = 2
• c₂ = - 5

Now,

$\displaystyle{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{2}{3}\:\:\:-\:-\:(\:1\:)}$

Now,

$\displaystyle{\sf\:\dfrac{b_1}{b_2}\:=\:\dfrac{3}{2}\:\:\:-\:-\:(\:2\:)}$

Now,

$\displaystyle{\sf\:\dfrac{c_1}{c_2}\:=\:-\:\dfrac{5}{5}}$

$\displaystyle{\implies\sf\:\dfrac{c_1}{c_2}\:=\:-\:1\:\:\:-\:-\:-\:(\:3\:)}$

From equations ( 1 ), ( 2 ) & ( 3 ),

$\displaystyle{\pink{\sf\:\dfrac{a_1}{a_2}\:\neq\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2}}}$

∴ The lines of the given pair of linear equations are intersecting.

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B)

The given linear equations are

x - y - 10 = 0 and 2x - 2y + 15 = 0

Now,

x - y - 10 = 0 - - ( 1 )

Comparing with ax + by + c = 0, we get,

• a₁ = 1
• b₁ = - 1
• c₁ = - 10

Now,

2x - 2y + 15 = 0 - - ( 2 )

Comparing with ax + by + c = 0, we get,

• a₂ = 2
• b₂ = - 2
• c₂ = 15

Now,

$\displaystyle{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{1}{2}\:\:\:-\:-\:-\:(\:1\:)}$

Now,

$\displaystyle{\sf\:\dfrac{b_1}{b_2}\:=\:\dfrac{\cancel{-}\:1}{\cancel{-}\:2}}$

$\displaystyle{\implies\sf\:\dfrac{b_1}{b_2}\:=\:\dfrac{1}{2}\:\:\:-\:-\:-\:(\:2\:)}$

Now,

$\displaystyle{\sf\:\dfrac{c_1}{c_2}\:=\:-\:\cancel{\dfrac{10}{15}}}$

$\displaystyle{\sf\:\dfrac{c_1}{c_2}\:=\:-\:\dfrac{2}{3}\:\:\:-\:-\:(\:3\:)}$

From equations ( 1 ), ( 2 ) & ( 3 ),

$\displaystyle{\underline{\boxed{\red{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2}}}}}$

∴ The lines of the given pair of linear equations are parallel.

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C)

The given linear equations are

2x + 5y + 2 = 0 and 4x - 10y + 4 = 0

Now,

2x + 5y + 2 = 0 - - ( 1 )

Comparing with ax + by + c = 0, we get,

• a₁ = 2
• b₁ = 5
• c₁ = 2

Now,

4x - 10y + 4 = 0 - - ( 2 )

Comparing with ax + by + c = 0, we get,

• a₂ = 4
• b₂ = - 10
• c₂ = 4

Now,

$\displaystyle{\sf\:\dfrac{a_1}{a_2}\:=\:\cancel{\dfrac{2}{4}}}$

$\displaystyle{\implies\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{1}{2}\:\:\:-\:-\:(\:1\:)}$

Now,

$\displaystyle{\sf\:\dfrac{b_1}{b_2}\:=\:-\:\cancel{\dfrac{5}{10}}}$

$\displaystyle{\implies\sf\:\dfrac{b_1}{b_2}\:=\:-\:\dfrac{1}{2}\:\:\:-\:-\:(\:2\:}$

Now,

$\displaystyle{\sf\:\dfrac{c_1}{c_2}\:=\:\cancel{\dfrac{2}{4}}}$

$\displaystyle{\implies\sf\:\dfrac{c_1}{c_2}\:=\:\dfrac{1}{2}\:\:\:-\:-\:(\:3\:)}$

From equations ( 1 ), ( 2 ) & ( 3 ),

$\displaystyle{\green{\sf\:\dfrac{a_1}{a_2}\:\neq\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2}}}$

∴ The lines of the given pair of linear equations are intersecting.

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D)

The given linear equations are

x - y = 10 and x + y = 14

Now,

x - y = 10

⇒ x - y - 10 = 0 - - ( 1 )

Comparing with ax + by + c = 0, we get,

• a₁ = 1
• b₁ = - 1
• c₁ = - 10

Now,

x + y = 14

⇒ x + y - 14 = 0 - - ( 2 )

Comparing with ax + by + c = 0, we get,

• a₂ = 1
• b₂ = 1
• c₂ = - 14

Now,

$\displaystyle{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{1}{1}}$

$\displaystyle{\implies\sf\:\dfrac{a_1}{a_2}\:=\:1\:\:\:-\:-\:-\:(\:1\:)}$

Now,

$\displaystyle{\sf\:\dfrac{b_1}{b_2}\:=\:-\:\dfrac{1}{1}}$

$\displaystyle{\implies\sf\:\dfrac{b_1}{b_2}\:=\:-\:1\:\:\:-\:-\:-\:(\:2\:)}$

Now,

$\displaystyle{\sf\:\dfrac{c_1}{c_2}\:=\:\dfrac{\cancel{-}\:10}{\cancel{-}\:14}}$

$\displaystyle{\implies\sf\:\dfrac{c_1}{c_2}\:=\:\cancel{\dfrac{10}{14}}}$

$\displaystyle{\implies\sf\:\dfrac{c_1}{c_2}\:=\:\dfrac{5}{7}\:\:\:-\:-\:-\:(\:3\:)}$

From equations ( 1 ), ( 2 ) & ( 3 ),

$\displaystyle{\blue{\sf\:\dfrac{a_1}{a_2}\:\neq\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2}}}$

∴ The lines of the given pair linear equations are intersecting.