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

##### 1 Answer

The solution is

#### 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

We start by assuming there is a point

#"-"x+2y=4#

#5x-3y=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

For example, if

#color(blue)(x=2y-4)#

by solving for *But*, if this is the same *substitute* this expression for

#" "5color(blue)x" "-3y=1#

#5(color(blue)(2y-4))-3y=1#

and we end up with an equation with just

#10y-20-3y=1#

#color(white)(10y-20-)7y=21#

#color(white)(10y-20-7)color(red)(y=3)#

So, this is the

#"-"x+2color(red)y" "=4#

#"-"x+2color(red)((3))=4#

#"-"x+6" "=4#

#"-"x" "="-"2#

#=>x=2#

That's it—we have found that the lines do cross, and the coordinates of the crossing point are

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