The curve C has the equation y=x^2+7. The line L has equation y=k(3x+1), where k is constant....help me

The curve C has the equation y=x^2+7. The line L has equation y=k(3x+1), where k is constant.

Show that the x co-ordinates of any points of intersection of the line L with the curve C satisfy the equation.

x^2-3kx+7-k=0

please explain in steps what this question is asking and how to answer it, thank you

If two curves intersect there x and y values should be same at intersection points.
hence, x*x+7 = k(3x+1)
x*x+7 - 3kx- k = 0
0r x*x - 3kx +7 - k = 0..... answer.

in order for the two graphs to have an intersection, their x and y values must equal at that point of intersection.

so the y in the first equation must equal the y in the other equation

well the y's are so graciously given to us in terms of x, we can equate x^2 + 7 with k(3x+1) and solve for x values.

if they are equated, you can subtract k(3x+1) from both sides and you will get the result desired