Question

3. State the type of the graph of the pair of linear equations: 3x - 5y = 11, 6x – 10y = 7
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Answer 1

Given :  3x - 5y = 11, 6x – 10y = 7

To Find :  type of the graph of the pair of linear equations

Solution:

Pair of linear equations

a₁x  +  b₁y + c₁  =  0

a₂x  +  b₂y + c₂  =  0

Consistent

if a₁/a₂ ≠ b₁/b₂   (unique solution  and lines intersects each others)

  a₁/a₂ = b₁/b₂ = c₁/c₂   (infinite solutions and line coincide each other )

Inconsistent

if  a₁/a₂ = b₁/b₂ ≠  c₁/c₂  ( No solution , lines are parallel to each other)

3x - 5y = 11,

6x – 10y = 7

a₁/a₂ = 3/6 = 1/2

b₁/b₂ = -5/-10 = 1/2

c₁/c₂  = 11/7

1/2 = 1/2 ≠   11/7

 a₁/a₂ = b₁/b₂ ≠  c₁/c₂  ( Inconsistent , No solution , lines are parallel to each other)

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Answer 2

SOLUTION

TO DETERMINE

State the type of the graph of the pair of linear equations: 3x - 5y = 11, 6x – 10y = 7

CONCEPT TO BE IMPLEMENTED

A pair of Straight Lines

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

is said to be parallel if

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

EVALUATION

Here the given pair of lines are

3x - 5y = 11, 6x – 10y = 7

Comparing with the lines

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

We have

$\displaystyle \sf{ a_1 = 3 \: , \: b_1 = - 5 , c_1= - 11 \: and \: \: a_2 = 6 \: , \: b_2 = - 10\: , \: \: c_2= - 7}$

Now

$\displaystyle \sf{ \: \frac{a_1}{a_2} = \frac{3}{6} \ = \: \frac{1}{2}}$

$\displaystyle \sf{ \: \frac{b_1}{b_2} = \frac{ - 5}{ - 10} = \frac{1}{2} }$

$\displaystyle \sf{ \frac{c_1}{c_2} = \frac{ - 11}{ - 7} = \frac{11}{7} }$

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

Hence the given pair of linear equations are parallel

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