Question

The pair of lines represented by the linear equations 3x + 2y = 7 and 4x + 8y - 11 = 0 are
a) perpendicular
(b) parallel
(c) intersecting
d) coincident ​

Answer 1

✤$\textsf{\large{\underline{Question}:-}}$

The pair of lines represented by the linear equations 3x + 2y = 7 and 4x + 8y - 11 = 0 are

(a) perpendicular

(b) parallel

(c) intersecting

(d) coincident

✤$\textsf{\large{\underline{Given}:-}}$

The equations are :-

$= > 3x + 2y = 7$

And,

$= > 4x + 8y - 11 = 0$

$= > 4x + 8y = 11$

✤$\textsf{\large{\underline{Solution}:-}}$

Here,

• $a_1$= 3
• $b_1$= 2
• $c_1$= 7

And,

• $a_2$= 4
• $b_2$= 8
• $c_2$= 11

Now,As we can see-

$a_1$/$a_2$≠$b_1$/$b_2$

• 3/4≠2/8

Therefore,the given system has a unique solution.

✤$\textsf{\large{\underline{Answer}:}}$

(c) intersecting

The pair of lines represented by the linear equations 3x + 2y = 7 and 4x + 8y - 11 = 0 are intersecting.

_______________________________

✤$\textsf{\large{\underline{Basic concepts}:-}}$

We have three conditions-

• An equation which can be put in the form ax+by+c=0 where a, b and c are real numbers,and a and b are not both zero, is called a linear equation in two variables x and y.
• Every solution of the equation is a point on the line representing it.

Solution of a pair of linear equations:-

$\textbf{Consistent system}$

1.Intersect at a point or unique solutions:-

$a_1$/$a_2$≠$b_1$/$b_2$

2.Infinite many solutions or coincident lines:-

$a_1$/$a_2$=$b_1$/$b_2$=$c_1$/$c_2$

$\textbf{Inconsistent system}$

3.Parallel lines or no solutions:-

$a_1$/$a_2$=$b_1$/$b_2$≠$c_1$/$c_2$

Answer 2

The pair of lines represented by the linear equations 3x + 2y = 7 and 4x + 8y - 11 = 0 are intersecting

Given:

• Pair of lines
• 3x + 2y = 7
• 4x + 8y - 11 = 0

To Find:

• Choose correct option:
• a) perpendicular
• (b) parallel
• (c) intersecting
• d) coincident ​

Solution:

Concept to be used:

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)

Step 1:

Rewrite the equations in required form:

3x + 2y - 7 = 0

4x + 8y - 11 = 0

Step 2:

Find the ratio of coefficients and compare

$\dfrac{3}{4} \neq \dfrac{2}{8}$

Hence Lines are intersecting

Correct option is c) intersecting

The pair of lines represented by the linear equations 3x + 2y = 7 and 4x + 8y - 11 = 0 are intersecting

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