# How do you find the point of intersection for 3x-y=4 and 6x+2y= -8?

Jul 4, 2015

Intersection point : (0,-4)

#### Explanation:

We want to find the point $A \left(X , Y\right)$ like :
$3 X - Y = 4$ and $6 X + 2 Y = - 8$

The word "intersection", here, is referring to functions :
A function is generally writing : $y = f \left(x\right)$

Then, we need to transform the two equations to something like :
"$y = \ldots$"
Let's define functions $f , g$, who are respectively representing equations $3 x - y = 4$ and $6 x + 2 y = - 8$

Function $f$ :
$3 x - y = 4 \iff 3 x = 4 + y \iff 3 x - 4 = y$
Then we have $f \left(x\right) = 3 x - 4$

Function $g$ :
$6 x + 2 y = - 8 \iff 2 y = - 8 - 6 x \iff y = - 4 - 3 x$
Then we have $g \left(x\right) = - 3 x - 4$

$A \left(X , Y\right)$ is an intersection point between $f$ and $g$ then :
$f \left(X\right) = Y$ and $g \left(X\right) = Y$
We can mark here $f \left(X\right) = g \left(X\right)$ and more :

$3 X - 4 = - 3 X - 4$
$\iff 3 X = - 3 X$ (we added 4 to each side)
$\iff 6 X = 0$
$\iff X = 0$

Then : $A \left(0 , Y\right)$ and $Y = f \left(0\right) = g \left(0\right) = - 4$

The coordinates of $A$ is $A \left(0 , - 4\right)$

We can check the result with a graph of the situation (Alone, this is not a proof !!)