# How do you solve the simultaneous equations 2x + 3y = 9 and 3x + 2y = 1?

Jul 30, 2015

Multiply each equation by a (different) constant so the coefficients of $x$ (or $y$) are identical then subtract one equation from the other to isolate a single variable.
$\left(x , y\right) = \left(- 3 , 5\right)$

#### Explanation:

[1]$\textcolor{w h i t e}{\text{XXXX}}$$2 x + 3 y = 9$
[2]$\textcolor{w h i t e}{\text{XXXX}}$$3 x + 2 y = 1$

Multiply [1] by 2 and [2] by 3 to get
[3]$\textcolor{w h i t e}{\text{XXXX}}$$4 x + 6 y = 18$
[4]$\textcolor{w h i t e}{\text{XXXX}}$$9 x + 6 y = 3$

Subtract [4] from [3]
[5]$\textcolor{w h i t e}{\text{XXXX}}$$- 5 x = 15$

Solve for $x$ by dividing [5] by $\left(- 5\right)$
[6]$\textcolor{w h i t e}{\text{XXXX}}$$x = - 3$

Substitute $\left(- 3\right)$ for $x$ in [1] to solve for $y$
[7]$\textcolor{w h i t e}{\text{XXXX}}$$2 \left(- 3\right) + 3 y = 9$

[8]$\textcolor{w h i t e}{\text{XXXX}}$$3 y = 15$

[9]$\textcolor{w h i t e}{\text{XXXX}}$$y = 5$