A line joins the pointis the point of intersection of 5x - 2y + 3 = 0 and 4x - 3y + 1 = 0 toof intersection of x = y and x = 3y + 4.find the equation of the line.
Accepted Solution
A:
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.