finding the gradient, coordinate and area of triangle

Question

line l1 has an equation $$y=3x+2$$ and line l2 has an equation $$3x+2y-8=0$$.

• find the gradient of line l2.The point of intersection of l1 and l2 is P.

• find the co-ordinate of P.

• the line l1 and l2 cross the line $$y=1$$ at the points A and B respectively. Find the area of the triangle ABP.

I can't answer these questions help me

Answer 1

For the line l2 you can rearrange it and write in the form y=ax+b where a the gradient and b the y intercept. Thus, you obtain that the gradient of the line is -2/3. Equating now l1 and l2: 3x+2=-3/2x+4, so x=4/9 and y=10/3. Similarly you can find that y=1 intersects l1 at x=-1/3 and l2 at x=2. Thus, the distance between these two points is d=2+1/3=7/3, which represents the base of the triangle. Its height can be found as 10/3-1 (the distance between the point of intersection of l1, l2 and the line y=1). Then apply Area = (1/2)*Base*Height to yield the Area. There are more direct ways of finding this area but they are more advanced from than the level of this question. It always helps to graph the lines though.

Answer 2

• Try to bring line l2 in the form of l1, like y=ax+b, then b is your gradient.
• When you have line l2 in this form, try to set y=y and replace the left side with l1 and the right side with l2. You will eventually get a value for x (that is the x-coordinate of P). Put that x in one of the equations for the lines and you will get the y coordinate of P.
• Try to make a drawing first, then look at the formular for the area of a triangle and imagine a way to get the needed lengths for it