Question

How do you solve the system `4x - 3y = 1` and `12x - 9y = 3` by substitution?

Answer 1

Dividing both sides of the second equation by `3` we get:

`4x-3y = 1`

which is the same as the first equation.

If we attempt to solve the system by substitution, then we will find that all of the terms in `x` and `y` cancel out, resulting in a true, but otherwise uninformative equation of rational numbers.

For example, if we take the first equation and add `3y` to both sides, then divide both sides by `4` we get:

`x = (3y + 1)/4`

Substitute this in the second equation:

`3 = 12x-9y = 12((3y+1)/4) -9y = 3(3y+1)-9y = 9y+3-9y = 3`

As a result, there are not enough constraints to determine a unique solution, but there are an infinite number of solutions.

These solutions are the points on the line which in slope intercept form is described by the equation:

`y = 4/3x + -1/3`