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

Aug 3, 2015

$\textcolor{red}{x = 3 , y = 2}$ and $\textcolor{red}{x = - 1 , y = - 2}$

#### Explanation:

One way is to use the method of elimination.

Step 1. Enter the equations.

[1] $x \left(y - 1\right) = 3$
[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 $x$.

$x \left(y - 1\right) = 3$

$x \left(x - 1 - 1\right) = 3$

$x \left(x - 2\right) = 3$

${x}^{2} - 2 x = 3$

${x}^{2} - 2 x - 3 = 0$

$\left(x - 3\right) \left(x + 1\right) = 0$

$x - 3 = 0$ 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: ($3 , 2$)

$3 \left(2 - 1\right) = 3$
$3 \left(1\right) = 3$
$3 = 3$

$2 = 3 - 1$
$2 = 2$

It checks!

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

$- 1 \left(- 2 - 1\right) = 3$
$- 1 \left(- 3\right) = 3$
$3 = 3$

$- 2 = - 1 - 1$
$- 2 = - 2$

It checks!

Our solutions are correct.