Step-by-step explanation:
We have got the lines :
[tex]y=ax+b\\y=cx+d[/tex]
Both lines intercept the x-axis in the point :
[tex]I = (i_{1} ,i_{2})[/tex]
In all point from x-axis the y-component is equal to 0.
[tex]I=(i_{1},o)[/tex]
We replace the I point in the lines equations:
[tex]0=a(i_{1})+b \\0=c(i_{1})+d[/tex]
From the first equation :
[tex]0=a(i_{1})+b \\-b=a(i_{1})\\i_{1}=\frac{-b}{a}[/tex]
From the second equation :
[tex]0=c(i_{1})+d\\ -d=c(i_{1})\\i_{1}=\frac{-d}{c}[/tex]
Then [tex]i_{1}=i_{1}[/tex]
Finally :
[tex]\frac{-b}{a}=\frac{-d}{c} \\\frac{b}{a}=\frac{d}{c} \\ad=bc[/tex]
y = ax + b and y = cx + d have the same x-intercept ⇔ad=bc
