Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given -2x - 9y = -25 and -4x - 9y = -23?

Answer 1

Solve one equation for x (or y), plug the result in the other equation, solve it for y (or x). You will get a unique solution, a statement that is always true or a statement that is always false.

Explanation:

You can solve one of the equations for `x` or `y` (take your pick, whichever looks easier) and plug it into the other equation.

Here, you might for example want to solve the first equation for `x`:
`-2x - 9y = - 25 <=> x = 25/2 - (9y)/2`

Afterwards, you plug `(25/2 - (9y)/2)` in the `x` of the second equation:

`-4 (25/2 - (9y)/2) - 9y = -23`

This equation only contains `y`, so you would want to solve it for `y`.

There are three possible cases:

1) you will be able to solve the equation and gain a unique solution for `y`.

This is the case here: simplifying the equation above you will get `y = 3` at the end. This means that the linear system has exactly one solution. Now, you just need to plug `y=3` in one of the original equations to find `x`.

2) simplifying the equation, `y` will "disappear" and you will get a generally true condition, like `1 = 1`.

This means that the linear system has infinitely many solutions.

3) simplifying the equation, `y` will "disappear and you will get a condition that is always false, like `1 = 0`.

This means that the linear equation system has no solutions at all.