Question

Help need correct answer ASAP

Two skaters are practicing at the same time on the same rink. A coordinate grid is superimposed on the ice. One skater follows
the path y = - 2x + 14, while the other skater follows the curve y = - 2x^2+ 14x.

Find all the points where they might collide if they
are not careful.

Answer 1

Answer: To find the points where the two skaters might collide, we need to solve the system of equations:

y = -2x + 14 (Equation 1)

y = -2x^2 + 14x (Equation 2)

We can substitute Equation 1 into Equation 2 to eliminate y and get:

-2x + 14 = -2x^2 + 14x

Simplifying, we get:

2x^2 - 16x + 14 = 0

Dividing both sides by 2, we get:

x^2 - 8x + 7 = 0

This quadratic equation factors as:

(x - 1)(x - 7) = 0

So the possible values of x where the skaters might collide are x = 1 and x = 7.

To find the corresponding y-values, we can plug each value of x into either Equation 1 or Equation 2. For x = 1:

y = -2(1) + 14 = 12 (using Equation 1)

y = -2(1)^2 + 14(1) = 12 (using Equation 2)

So the skaters might collide at the point (1, 12).

For x = 7:

y = -2(7) + 14 = 0 (using Equation 1)

y = -2(7)^2 + 14(7) = 0 (using Equation 2)

So the skaters might collide at the point (7, 0).

Therefore, the two skaters might collide at points (1, 12) and (7, 0).

Explanation: