# What is the solution of the system of equations: -6x+2y=-8 ?

Feb 2, 2017

$\left(x , y\right) \in \left\{\left(t , 3 t - 4\right) : t \in \mathbb{R}\right\}$

#### Explanation:

This system of equations only has one linear equation in two unknowns and hence an infinite number of solutions.

The solutions lie along the line described by the given equation:

$- 6 x + 2 y = - 8$

We can rearrange this equation to express $x$ in terms of $y$ or $y$ in terms of $x$ as follows:

Divide both sides of the equation by $2$ to get:

$- 3 x + y = - 4$

Add $3 x$ to both sides to get:

$\textcolor{b l u e}{y = 3 x - 4}$

For any value of $x$ this gives us the corresponding value of $y$.

This formula is in the form:

$y = m x + c$

known as slope intercept format, where $m = 3$ is the slope of the line and $c = - 4$ is the $y$ intercept.

If we add $4$ to both sides and transpose we get:

$3 x = y + 4$

Then dividing both sides by $3$ we get:

$\textcolor{b l u e}{x = \frac{1}{3} y + \frac{4}{3}}$

For any given $y$, this formula gives us the corresponding value of $x$.

Alternatively, we can use the previous slope intercept format equation to derive a parametric representation of the line as:

$\left(t , 3 t - 4\right)$

where $t \in \mathbb{R}$

So we can express the solution space of the original system of equation(s) as:

$\textcolor{b l u e}{\left(x , y\right) \in \left\{\left(t , 3 t - 4\right) : t \in \mathbb{R}\right\}}$

graph{y=3x-4 [-9.42, 10.58, -5.72, 4.28]}