What is the solution to {:(–x+2y=4),(5x-3y=1):}?

Jan 13, 2017

The solution is $\left(x , y\right) = \left(2 , 3\right)$.

Explanation:

Each of these equations represents a line in 2D space. As with any pair of lines, they may cross, they may be parallel, or they may be the same line. Solving a pair of equations simultaneously means finding the $\left(x , y\right)$ point where the lines cross (if it exists).

We start by assuming there is a point $\left(x , y\right)$ that works for both equations

$\text{-} x + 2 y = 4$
$5 x - 3 y = 1$

If this is true, then we can rearrange each equation and combine the two equations together to help us narrow in on the coordinates of the $\left(x , y\right)$ point.

For example, if $\text{-} x + 2 y = 4$, then we have

$\textcolor{b l u e}{x = 2 y - 4}$

by solving for $x$. But, if this is the same $x$ that works for the other equation, we can substitute this expression for $x$ into the other equation like this:

$\text{ "5color(blue)x" } - 3 y = 1$
$5 \left(\textcolor{b l u e}{2 y - 4}\right) - 3 y = 1$

and we end up with an equation with just $y$. Thus, we can solve for $y$:

$10 y - 20 - 3 y = 1$
$\textcolor{w h i t e}{10 y - 20 -} 7 y = 21$
$\textcolor{w h i t e}{10 y - 20 - 7} \textcolor{red}{y = 3}$

So, this is the $y$-coordinate of the point that works for both lines. With this, we can now find the matching $x$-coordinate, by plugging in this value for $y$ into either of our starting equations:

$\text{-"x+2color(red)y" } = 4$
$\text{-} x + 2 \textcolor{red}{\left(3\right)} = 4$
$\text{-"x+6" } = 4$
$\text{-"x" "="-} 2$

$\implies x = 2$

That's it—we have found that the lines do cross, and the coordinates of the crossing point are $\left(x , y\right) = \left(2 , 3\right)$:

graph{(-x+2y-4)(5x-3y-1)=0 [-10, 10, -2, 8]}