Question

How do you solve the simultaneous equations `3x-4y=11` and `5x+6y=12`?

Answer 1

`x=3`; `y=-1/2`

Explanation:

You can solve this system of equations by using the multiplication method.

The first thing that you need to do is pick a variable to eliminate first, then figure out the least common multiple (LCM) of its coefficients.

Let's say that you want to eliminate `x` and solve for `y` first. The two coefficients of `x` are `3` and `5`, which means that they're LCM will be equal to `15`.

So, multiply the first equation by `5` and the second equation by `-3` to get

`5 * (3x - 4y) = 5 * 11`

`15x - 20y = 55`

and

`(-3) * (5x + 6y)= -3 * 12`

`-15x - 18y = -36`

Add the left side and the right side of these two equations separately to get

`color(red)(cancel(color(black)(15x))) - 20y - color(red)(cancel(color(black)(15x))) - 18y = 55 - 36`

`-38y = 19 => y = 19/(-38) = color(green)(-1/2)`

Now use this value of `y` in one of the two equations to determine the value of `x`.

`3x - 4(-1/2) = 11`

`3x + 2 = 11 => x = (11-2)/3 = color(green)(3)`

The solutions to this system of equations are

`{(x=3), (y=-1/2) :}`