# How do solve the following linear system?:  7x-2y=8 , 3x – 4y = 7 ?

Dec 24, 2015

By substitution or by elimination

#### Explanation:

you are given a 2 equation and you need to find both x and y

$\left\{\begin{matrix}7 x - 2 y = 8 \\ 3 x - 4 y = 7\end{matrix}\right.$

there are two solutions and they have both the same answer. you will just choose what is easier.

By substitution
first let's find the value of x in first equation

$7 x - 2 y = 8$

$7 x = 8 + 2 y \to$ transpose $2 y$ to other side of equation to leave $7 x$

$x = \frac{8 + 2 y}{7} \to$ divide both side by $7$

then we have the x value

substitute your answer (the value of $x$) to next equation

$3 \cdot \left(\frac{8 + 2 y}{7}\right) - 4 y = 7$

$\frac{24 + 6 y}{7} - 4 y = 7$

$\frac{24 + 6 y}{7} - 4 y = 7 | \cdot \left(7\right)$

$24 + 6 y - 28 y = 49$

$- 22 y = 49 - 24$

$- 22 y = 25$

$y = - \frac{25}{22}$

To find the value of $x$ just substitute the first equation by the value of $y$.

BY ELIMINATION

$\left\{\begin{matrix}7 x - 2 y = 8 \\ 3 x - 4 y = 7\end{matrix}\right.$

Think of how to eliminate any one of the variables.

i think that if we multiply the first equation by $- 2$, $y$ will be cancelled.

$\left\{\begin{matrix}\left(7 x - 2 y = 8\right) | \cdot \left(- 2\right) \\ 3 x - 4 y = 7\end{matrix}\right.$

$- 14 x + 4 y = - 16$
$\text{ } 3 x - 4 y = 7$

Add the two equations to get

$- 14 x + \cancel{4 y} + 3 x - \cancel{4 y} = - 16 + 7$

$- 11 x = - 9 \to$ divide both sides by $- 11$

$x = \frac{9}{11}$

To know the value of $y$, substitute the value of $x$ in one of the equation.

$3 x - 4 y = 7$

$3 \cdot \left(\frac{9}{11}\right) - 4 y = 7$

$\frac{27}{11} - 4 y = 7$

$- 4 y = 7 - \frac{27}{11}$

$- 4 y = \frac{50}{11} \to$ divide both sides by $- 4$

$y = - \frac{25}{22}$