# Two skaters are at the same time on the same rink. One skater follows the path y=-2x^2+18x while the other skater follows a straight path that begins at (1, 30) and ends at (10, 12). How do you write a system of equations to model the situation?

## Is it possible the two skaters might collide?

Feb 19, 2016

Since we already have the quadratic equation (a.k.a the first equation), all we must find is the linear equation.

#### Explanation:

First, find the slope using the formula $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$, where m is slope and $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ are points on the graph of the function.

$m = \frac{30 - 12}{1 - 10}$

$m = \frac{18}{-} 9$

$m = - 2$

Now, plugging this into point slope form. Note: I used the point (1,30) but either point would result in the same answer.

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$y - 30 = - 2 \left(x - 1\right)$

$y = - 2 x + 2 + 30$

$y = - 2 x + 32$

In slope intercept form, with y isolated, the term with x as its coefficient would be the slope and the constant term would be the y intercept.

You would be best off solving the system by graphing, because the line has start and end points that are not written directly in the equation. First graph the function. Then, erase all parts that are outside your start and end points. Finish by graphing the parabola.