Question

'For all real values of 'c', the pair of equations x – 2y = 8 and 5x-10y=c have an unique solution.
Justify whether it is true or false.​

Answer 1

Answer:

false

Step-by-step explanation:

this type of equation is :

a1/a2=b1/b2=c1/c2

coefficient of x is 1 and 5

coefficient of y is -2 and -10

so 1/5 = -2/-10(i.e 1/5) is not equal to 8/1

Answer 2

QUESTION:

• "For C ∈ R, the pair of linear equations x - 2y = 8 and 5x - 10y = C have unique solution." True or False.

ANSWER:

• False

Given:

• x - 2y = 8
• 5x - 10y = C
• C ∈ R

To Do:

• True or False

Understanding Concept:

Let there be a pair of linear equations,

$\hookrightarrow a_1x+b_1y=c_1\:\:\:\&\:\:\:a_2x+b_2y=c_2$

We know that, for a pair of linear equations, the ratio of corresponding coefficients of variables and constant terms gives following results:

$\longrightarrow If,\:\:\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

Then, 1 solution exist and graphically they intersect with each other.

$\longrightarrow If,\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

Then, infinite solutions exist and graphically they coincide with each other.

$\longrightarrow If,\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$

Then, 0 solutions exist and graphically they are parallel to each other.

Solution:

We are given that,

$\longrightarrow x-2y=8$

And,

$\longrightarrow5x-10y=C$

Here,

$\hookrightarrow a_1=1,\:a_2=5,\:b_1=-2,\:b_2=-10,\:c_1=8,\:c_2=C$

Now we'll compare the ratios,

$\implies\dfrac{a_1}{a_2}=\dfrac{1}{5}- - - -(1)$

$\implies\dfrac{b_1}{b_2}=\dfrac{-2}{-10}=\dfrac{1}{5}- - - -(2)$

For this question, we need ratios of coefficients of x and y only.

This is so because, we need to find whether the equations will give Unique Solution or not.

So, for the equations to have unique solution,

$\hookrightarrow\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

Substituting values from (1) and (2),

$\implies\dfrac{1}{5}\neq\dfrac{1}{5}$

As, the above statement is incorrect,

Hence, the equations don't have Unique Solution.

Therefore, the statement that "For C ∈ R, the equations have unique solution" is FALSE.