and are points on the parabola such that and intersect on the -axis. Determine the -coordinate of in terms of the -coordinates of and , which are and respectively.
Solution:Start out by graphing , and labeling lines and , such that all points are on the line . We label the point , the point , the point , and the point . The lines intersect at the y-axis, which we label to be point . We start by finding the equation is on. We put the line into the slope-intercept form , where is the slope and is the y-intercept* . Since the line intersects the at the point , we know the y-intercept is . The slope of the line is . Therefore we have . Substituting the point into the equation gives us . We could also substitute into the equation to give us as well. We now find the equation is on. Again, put the line into slope-intercept form , where is slope, and is y-intercept*. The lines and intersect at the point , therefore the y-intercept of is . The slope of is . Therefore our new equation is . Substituting either points or give or , where both can be simplified two after expanding and subtract in the first equation, and in the second equation to give . We are now left with the two equations and . Subtracting the two equations results in *(note: I changed the slope-intercept form from to to avoid any confusion with point )