Intersection of the first two lines:
[tex]\begin{cases}5x - 2y + 3 = 0\\4x - 3y + 1 = 0\end{cases}[/tex]
Multiply the first equation by 4 and the second by 5:
[tex]\begin{cases}20x - 8y + 12 = 0\\20x - 15y + 5 = 0\end{cases}[/tex]
Subtract the two equations:
[tex](20x - 8y + 12)-(20x - 15y + 5)=0 \iff 7y+7=0 \iff y=-1[/tex]
Plug this value for y in one of the equation, for example the first:
[tex]5x - 2\cdot (-1) + 3 = 0\iff 5x+5=0 \iff x=-1[/tex]
So, the first point of intersection is [tex](-1,-1)[/tex]
We can find the intersection of the other two lines in the same way: we start with
[tex]\begin{cases}x=y\\x=3y+4\end{cases}[/tex]
Use the fact that x and y are the same to rewrite the second equation as
[tex]x=3x+4 \iff 2x=-4 \iff x=-2[/tex]
And since x and y are the same, the second point is [tex](-2, -2)[/tex]
So, we're looking for a line passing through [tex](-1,-1)[/tex] and [tex](-2, -2)[/tex]. We may use the formula to find the equation of a line knowing two of its points, but in this case it is very clear that both points have the same coordinates, so the line must be [tex]y=x[/tex]
In the attached figure, line [tex]5x - 2y + 3 = 0[/tex] is light green, line [tex]4x - 3y + 1 = 0[/tex] is dark green, and their intersection is point A.
Simiarly, line [tex]x=y[/tex] is red, line [tex]x = 3y + 4[/tex] is orange, and their intersection is B.
As you can see, the line connecting A and B is the red line itself.
