ind the solution of the system of equations{ x − 5y = −277x + 3y = 39[4]4. Consider the function y = −2x2 + 2x + 24.(a) Find its y-intercept.(b) Find its x-intercepts.(c) Sketch its graph, labelling the axes and intercepts clearly.(d) Hence solve the inequality −2x2 + 2x + 24 ≥ 0.

Sure, let's solve this step by step:

1. Solve the system of equations:

The system of equations is: x - 5y = -27 7x + 3y = 39

We can solve this system by substitution or elimination. Let's use substitution:

First, solve the first equation for x: x = 5y - 27

Then, substitute x into the second equation: 7(5y - 27) + 3y = 39 35y - 189 + 3y = 39 38y = 228 y = 228 / 38 y = 6

Substitute y = 6 into the first equation: x - 5*6 = -27 x - 30 = -27 x = -27 + 30 x = 3

So, the solution to the system of equations is x = 3, y = 6.

2. Consider the function y = -2x^2 + 2x + 24:

(a) Find its y-intercept: The y-intercept is the value of y when x = 0. So, substitute x = 0 into the equation: y = -20^2 + 20 + 24 y = 24

(b) Find its x-intercepts: The x-intercepts are the values of x when y = 0. So, set y = 0 and solve for x: 0 = -2x^2 + 2x + 24 2x^2 - 2x - 24 = 0 x^2 - x - 12 = 0 (x - 4)(x + 3) = 0 So, x = 4 and x = -3 are the x-intercepts.

(c) Sketch its graph: I'm sorry, as a text-based AI, I can't sketch graphs. However, you can plot the function y = -2x^2 + 2x + 24 and mark the intercepts (0,24), (4,0), and (-3,0).

(d) Solve the inequality -2x^2 + 2x + 24 ≥ 0: This inequality is satisfied when y = -2x^2 + 2x + 24 is greater than or equal to 0. From the x-intercepts found in part (b), we know that this function is 0 at x = 4 and x = -3. Since the coefficient of x^2 is negative, the parabola opens downwards, meaning it is above the x-axis (i.e., y is greater than 0) between the x-intercepts. So, the solution to the inequality is -3 ≤ x ≤ 4.