# Question d5b6d

Sep 29, 2017

$\left(5 , 1\right)$

$\left(- 11 , \frac{67}{3}\right)$

#### Explanation:

$4 x + 3 y = 23 - - - \left(1\right)$

$2 {x}^{2} - 9 y = 41 - - - \left(2\right)$

normally with these we would rearrange one eqn for one variable and substitute into eh other eqn.

Here, however we can quickly eliminate one of the variables as follows

$\times \left(1\right) \text{ by } 3$

$12 x + 9 y = 69 - - - \left(1 a\right)$

$2 {x}^{2} - 9 y = 41 - - - \left(2\right)$

$2 {x}^{2} + 12 x = 110$

$\implies {x}^{2} + 6 x - 55 = 0$

the rearranging is left as an exercise for the reader

$\left(x + 11\right) \left(x - 5\right) = 0$

$\implies x = 5 \text{ or } x = - 11$

$x = 5$

using $\left(1\right)$

$4 \times 5 + 3 y = 23$

$\implies 3 y = 3$

:.y=1

one soln$\left(5 , 1\right)$

$x = - 11$

$4 x + 3 y = 23$
-44+3y=23#

$3 y = 67$

$\therefore y = \frac{67}{3}$

other soln

$\left(- 11 , \frac{67}{3}\right)$

Sep 29, 2017

$x = 5 , - 11$
$y = 1 , - \frac{67}{3}$

#### Explanation:

$4 x + 3 y = 23$
$3 y = - 4 x + 23$
$9 y = 3 \left(- 4 x + 23\right) = - 12 x + 69$
Substituting the value of 9y in the second equation,
$2 {x}^{2} + 12 x - 69 = 41$
$2 {x}^{2} + 12 x - 110 = 0$
$2 {x}^{2} - 10 x + 22 x - 110 = 0$
$2 x \left(x - 5\right) + 22 \left(x - 5\right) = 0$
$\left(2 x + 22\right) \left(x - 5\right) = 0$
$x = 5 , - 11$
$4 x + 3 y = 23$
$3 y = - 4 x + 23$
When $x = 5$, $3 y = - 20 + 23 = 3$ or $y = 1$
When $x = - 11$, $3 y = 44 + 23 = 67$; $y = \frac{67}{3}$