Question

Which of these describes the system of linear equations below?

3x-2y=7
6x-4y=14

a) the system has no solutions.
b) the system has infinitely many solutions.
c) the ration of the-coordinate to the y-coordinate of the only solution is 2 : 1.
d) the difference between the x-coordinate and y-coordinate of the only solution is one.

Answer 1

Given:

The system of equations is

$3x-2y=7$

$6x-4y=14$

To find:

The correct statement for the given system of equations.

Solution:

On comparing the given equations and general form of linear equation, i.e., $ax+by+c=0$, we get

$a_1=3,b_1=-2,c_1=-7$

$a_2=6,b_2=-4,c_2=-14$

Here,

$\dfrac{a_1}{a_2}=\dfrac{3}{6}=\dfrac{1}{2}$

$\dfrac{b_1}{b_2}=\dfrac{-2}{-4}=\dfrac{1}{2}$

$\dfrac{c_1}{c_2}=\dfrac{-7}{-14}=\dfrac{1}{2}$

Since, $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$, therefore, the two equations are equivalent and the system has infinitely many solutions.

Hence, the correct option is b.