Question

How do you solve the simultaneous equations `x(y-1)=3` and `y=x-1`?

Answer 1

`color(red)(x=3,y=2)` and `color(red)(x=-1,y=-2)`

Explanation:

One way is to use the method of elimination.

Step 1. Enter the equations.

[1] `x(y-1)=3`
[2] `y=x-1`

Step 2. Solve for one variable in terms of the other.

[2] `y=x-1`

Since this is already done for us, we can go on to the next step.

Step 3. Substitute Equation 2 in Equation 1 and solve for `x`.

`x(y-1)=3`

`x(x-1-1)=3`

`x(x-2)=3`

`x^2-2x=3`

`x^2-2x-3=0`

`(x-3)(x+1)=0`

`x-3=0` and `x+1=0`

`x=3` and `x=-1`

Step 4. Substitute each value of `x` in Equation 2

If `x=3`,
`y=3-1`
`y=2`

If `x=-1`,
`y=-1-1`
`y=-2`

Solutions: `x=3,y=2` and `x=-1,y=-2`

Check: Substitute the values of `x` and `y` in Equations 1 and 2.

Check No. 1: (`3,2`)

`3(2-1)=3`
`3(1) =3`
`3=3`

`2=3-1`
`2=2`

It checks!

Check No. 2: (`-1,-2`)

`-1(-2-1)=3`
`-1(-3)=3`
`3=3`

`-2=-1-1`
`-2=-2`

It checks!

Our solutions are correct.