Question

The pair of linear equations 3x - 5y = 7 and 6x - 10y = 7 has
(a) one solution
(b) infinitely many solutions
(c) two solutions
(d) no solution​

Answer 1

Given:

Two linear equations:

$3x - 5y = 7\\ 6x - 10y = 7$

To find:

The number of solutions for the above equations = ?

Solution:

The given equations are of straight lines and straight line can have one(the intersection point) or infinite many (in case of identical lines) or no solutions(in case of parallel lines).

Let us learn the conditions one by one:

1. One solution: If the slopes of two given lines are different, then the equations will have exactly one solution.

2. Infinite many solutions: If the equations of two lines come out to be same, the two lines will be co-incident and will have infinite many solutions.

3. No Solutions: If the slopes of two lines are equal and y intercept is not equal, that means the lines will be parallel to each other and will not intersect with each other, hence no solutions.

Let us convert the given lines to the the slope-intercept form:

i.e. $y = mx+c$

where (x,y) are the coordinates of points on line.

m is the slope and

c is the y intercept.

$3x - 5y = 7\\\Rightarrow 5y=3x-7\\\Rightarrow y = \dfrac{3}{5}x-\dfrac{7}{5} ..... (1)\\\\6x - 10y = 7\\\Rightarrow 10y =6x-7\\\Rightarrow y =\dfrac{6}{10}x-\dfrac{7}{10}\\\Rightarrow y =\dfrac{3}{5}x-\dfrac{7}{10} ...... (2)$

Comparing the equations with the standard equation $y = mx+c$:

The slopes

$m_1 =m_2 = \dfrac{3}{5}$

y intercepts:

$c_1 = -\dfrac{7}{5}\\c_2 = -\dfrac{7}{10}$

Please refer to the graphical representation of the given lines in the attached figure.

Now, the slopes are equal but y intercepts are not same

$\therefore$ the lines are parallel to each other.

Hence, they have No solution.

The option "(d) No solution " is correct.

Answer 2

The pair of linear equations 3x - 5y = 7 and 6x - 10y = 7 has no solution.

Step-by-step explanation:

We are given the following pair of linear equations below;

$3x-5y=7$  and  $6x-10y=7$

There are basically three types of solutions for the pair of linear equations, i.e;

Unique solution, Infinitely many solutions and no solution.

The conditions for each of the solutions is given by;

• If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, then there is a unique solution for the pair of linear equations.

• If $\frac{a_1}{a_2} = \frac{b_1}{b_2}=\frac{c_1}{c_2}$, then there are infinitely many solutions for the pair of linear equations.

• If $\frac{a_1}{a_2} = \frac{b_1}{b_2}\neq \frac{c_1}{c_2}$, then there is no solution for the pair of linear equations.

So, according to our question;

$a_1=3, b_1=-5 \text{ and } c_1=7$

$a_2=6, b_2=-10 \text{ and } c_2=7$

SO, $\frac{a_1}{a_2} =\frac{3}{6} = \frac{1}{2}$

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

$\frac{c_1}{c_2} =\frac{7}{7} = 1$

Here, the third condition is satisfying that;  $\frac{a_1}{a_2} = \frac{b_1}{b_2}\neq \frac{c_1}{c_2}$ which means $\frac{1}{2}= \frac{1}{2}\neq 1$.

Therefore, the given pair of linear equations has no solution.