Check whether the following lines g,h and i run through a common point.
g(x)=x+1;h:2y+x+4=0;3y−5x=7
For this task you need the following basic knowledge: Linear function
First bring all lines equations into the general form y=mx+t.
Line g:
g(x)=x+1
⇔y=x+1
Line h:
2y+x+4=0
⇔2y=−x−4
⇔y=−12x−2
Line i:
3y−5x=7
⇔3y=5x+7
⇔y=53x+73
Set the respective right sides equal.
Get all x-terms on one side.
Conclude
Plug into g to get the y-coordinate.
The intersection point is Sgh(−2|−1).
Conclude.
The intersection point is Sgi(−2|−1).
Since g intersects with h and with i at the same point, h and i also intersect at this point.
The lines therefore all run through the common point (−2|−1).
g(x)=16x+32;h(x)=−23x+2;i:2x−y=3
g(x)=16x+32
⇔y=16x+32
h(x)=−23x+2
⇔y=−23x+2
2x−y=3
⇔y=2x−3
The intersection point is Sgh(35|85).
The intersection point is Sgi(2711|2111).
Thus the line g intersects the line h at a different point than the line i. So the lines do not run through a common point.