# How do you solve the following linear system: -9 y + 2x = 2 , 5x - y = 3?

May 30, 2017

See a solution process below:

#### Explanation:

Step 1) Solve the second equation for $y$:

$5 x - y = 3$

$- \textcolor{red}{3} + 5 x - y + \textcolor{b l u e}{y} = - \textcolor{red}{3} + 3 + \textcolor{b l u e}{y}$

$- 3 + 5 x - 0 = 0 + \textcolor{b l u e}{y}$

$- 3 + 5 x = y$

$y = - 3 + 5 x$

Step 2) Substitute $\left(- 3 + 5 x\right)$ for $y$ in the first equation and solve for $x$:

$- 9 y + 2 x = 2$ becomes:

$- 9 \left(- 3 + 5 x\right) + 2 x = 2$

$\left(- 9 \times - 3\right) + \left(- 9 \times 5 x\right) + 2 x = 2$

$27 - 45 x + 2 x = 2$

$27 + \left(- 45 + 2\right) x = 2$

$27 - 43 x = 2$

$- \textcolor{red}{27} + 27 - 43 x = - \textcolor{red}{27} + 2$

$0 - 43 x = - 25$

$- 43 x = - 25$

$\frac{- 43 x}{\textcolor{red}{- 43}} = \frac{- 25}{\textcolor{red}{- 43}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 43}}} x}{\cancel{\textcolor{red}{- 43}}} = \frac{25}{43}$

$x = \frac{25}{43}$

Step 3) Substitute $\frac{25}{43}$ for $x$ in the solution to the second equation at the end of Step 1 and calculate $y$:

$y = - 3 + 5 x$ becomes:

$y = - 3 + \left(5 \times \frac{25}{43}\right)$

$y = - 3 + \frac{125}{43}$

$y = \left(- 3 \times \frac{43}{43}\right) + \frac{125}{43}$

$y = - \frac{129}{43} + \frac{125}{43}$

$y = - \frac{4}{43}$

The solution is: $x = \frac{25}{43}$ and $y = - \frac{4}{43}$ or $\left(\frac{25}{43} , - \frac{4}{43}\right)$

May 30, 2017

$x = \frac{25}{43} , y = - \frac{4}{43.}$

#### Explanation:

From the second eqn., $5 x - 3 = y .$

Subst.ing this value of $y$ in the first eqn., we get,

$- 9 \left(5 x - 3\right) + 2 x = 2.$

$\therefore - 45 x + 27 + 2 x = 2.$

$\therefore - 43 x = 2 - 27 = - 25.$

$\therefore x = \frac{25}{43.}$

$\therefore y = 5 x - 3 = 5 \left(\frac{25}{43}\right) - 3 = \frac{125}{43} - 3 = \frac{125 - 129}{43.}$

;. y=-4/43.

Hence, the Soln. $x = \frac{25}{43} , y = - \frac{4}{43.}$