Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given x - 4y = 2 and 2x - 8y = 5?

Answer 1

No solution

Explanation:

`x - 4y = 2`

`2x - 8y = 5`

If we factor out 2 from the second equation, we will have the same coefficients as the first equation

`=> 2(x - 4y) = 5`

What does this mean? it means the two lines are parallel.
Either there are infinitely many solutions (i.e. the lines coincide) or there is none (i.e. non-trivial case).

How do we determine if the lines coincide? We use the right-hand side of the equation.

If we divide both sides of the second equation by 2, we will get

`2(x - 4y) = 5`

`=> x - 4y = 5/2`

Since the right-hand side of the equation of the second equation is not equal to that of the first equation, we can conclude that the lines do not coincide. Hence, there is no solution