# How do you write a system of equations that will have (3,2) as a solution?

Mar 28, 2015

There are many pairs of equations that have $\left(3 , 2\right)$ as a solution.

Probably the simplest pair is
$x = 3$
$y = 2$

A less trivial pair of linear equations could be generated using the slope-point formula for a straight line:
$\frac{y - 2}{x - 3} = m$ where $m$ is an arbitrary slope
By picking two different values for $m$ (say $2$ and $5$)
we would have
$\frac{y - 2}{x - 3} = 2$
$\rightarrow y = 2 x - 4$
and
$\frac{y - 2}{x - 3} = 5$
$\rightarrow y = 5 x - 13$

If you want one of the equations to be non-linear, for example a quadratic of the form
$y = {x}^{2} - 7 x + c$ for some constant value $c$
simply plug in $\left(3 , 2\right)$ for $\left(x , y\right)$ to solve for $c$
$\rightarrow c = 14$
so
$y = {x}^{2} - 7 x + 14$
combined with any one of the previous (linear) equations would have a solution of $\left(3 , 2\right)$ (although it might not be the only solution).