Question

Which statement must be true about the system below? The system has exactly one solution. The system has exactly two solutions. The system has no solutions. The system has infinitely many solutions.

Answer 1

Answer: The system has infinitely many solutions.

Explanation:

The given system of equations :

`2h+8k=6\\\\-5h-20k=-15`

Let
`a_1=2\ ; b_1=8\ ;\ c_1=5`

`a_2=-5\ ;\ b_2=-20\ ;\ c_2=-15`

Now,
`(a_1)/(a_2)=(2)/(-5)=(-2)/(5)`

`(b_1)/(b_2)=(8)/(-20)=(-2)/(5)` [Divide numerator and denominator by 4]

`(c_1)/(c_2)=(6)/(-15)=(-2)/(5)` [Divide numerator and denominator by 3]

⇒
`(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)=(-2)/(5)`

It means the system is linearly dependent and so it has infinitely many solutions.

Answer 2

The system has infinitely solutions

Explanation:

The system shown in the attached picture can be simplify by performing natural divisions

`h+4k=4\\h+4k=3`

Since the equations are equal in terms, we have a system where one equation is a contradiction of the other

On one side it says h+4k=4 and the other says h+4k=3

This system has more variables than equations, it has infinitely solutions