Question

pair of equations 3x - 5y = 7 and - 6x + 10y = 7 have
unique solution b.infinitely many solutions C. no solution d. two solutions
​

Answer 1

ANSWER:

Given:

• Pair of equations, 3x - 5y = 7 and -6x + 10y = 7

To Find:

• No. of Solution

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:

$1):\longrightarrow If,\:\:\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

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

$2):\longrightarrow If,\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

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

$3):\longrightarrow If,\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$

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

Solution:

We are given that,

$:\longrightarrow 3x-5y=7$

And,

$:\longrightarrow-6x+10y=7$

Here,

$:\hookrightarrow a_1=3,\:a_2=-6,\:b_1=-5,\:b_2=10,\:c_1=7,\:c_2=7$

Now we'll compare the ratios,

$:\implies\dfrac{a_1}{a_2}=\dfrac{3}{-6}=\dfrac{-1}{2}- - - -(1)\\\\:\implies\dfrac{b_1}{b_2}=\dfrac{-5}{10}=\dfrac{-1}{2}- - - -(2)\\\\:\implies\dfrac{c_1}{c_2}=\dfrac{7}{7}=1- - - -(3)$

Now, we compare the ratios,

[from (1), (2) & (3),]

$:\implies\dfrac{-1}{2}=\dfrac{-1}{2}\neq1\\\\:\implies\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{_1}{c_2}$

We can see that, the above relation is the last(3rd) case where ratio of coefficients of x and y are equal not equal to the ratio of constants.

So,

The given pair of equations have [c)] No Solution.