# Question #e874b

Sep 18, 2017

x=4 and y=1

#### Explanation:

(1) $3 x - 5 y = 7$
(2) $y - x = - 3$

Multiplying equations (2) with 5,we have

(3)$5 y - 5 x = - 15$
Adding equations (1) and (3),
$\left(3 x - 5 y\right) + \left(5 y - 5 x\right) = \left(7\right) - \left(15\right)$
$\left(3 x - 5 x\right) + \left(- 5 y + 5 y\right) = - 8$
$- 2 x = - 8$
$x = 4$

Substituting $x = 4$ in equation (2),
$y - \left(4\right) = - 3$
$y = - 3 + 4$
$y = 1$

For graphical approach,
a linear equation of the form $a x + b y + c = 0$ is a straight line which can be plotted on graph plotting the points $\left(- \frac{c}{a} , 0\right)$ and $\left(0 , - \frac{c}{b}\right)$ when $a \ne 0 \mathmr{and} b \ne 0$ and joining them through a straight line.
The solution for two linear equations can be found by plotting the straight lines corresponding to the two equations and finding the intersection point.
If the intersection is a unique point,the solution is unique.
If the lines do not intersect,there is no solution.
If the lines coincide,the solution is infinite points forming that line.

For given question,the two straight lines $3 x - 5 y = 7$ can be plotted as follow
graph{3x-5y=7 [-10, 10, -5, 5]}

and the line $y - x = - 3$ can be plotted as follow graph{y-x=-3 [-10, 10, -5, 5]}

On plotting the two lines on same graph,the intersection point turns out to be $\left(4 , 1\right)$ which is the unique solution to our equations