Question

select all the correct answers. determine the solutions for the system of equations shown below. $y = -2x + \frac{21}{4}$ $y = x^2 - x - \frac{3}{4}$ $\quad (2, \frac{5}{4})$ $\quad (-1, \frac{29}{4})$ $\quad (2, \frac{37}{4})$ $\quad (-3, \frac{3}{4})$ $\quad (-3, \frac{45}{4})$ $\quad (-1, \frac{5}{4})$

Explanation:

Step1: Set equations equal to each other

Since $y = -2x + \frac{21}{4}$ and $y = x^2 - x - \frac{3}{4}$, we equate the right-hand sides:
$$-2x + \frac{21}{4} = x^2 - x - \frac{3}{4}$$

Step2: Eliminate fractions, simplify equation

Multiply all terms by 4 to clear denominators:
$$-8x + 21 = 4x^2 - 4x - 3$$
Rearrange into standard quadratic form $ax^2+bx+c=0$:
$$4x^2 + 4x - 24 = 0$$
Divide by 4 to simplify:
$$x^2 + x - 6 = 0$$

Step3: Factor the quadratic

Factor the simplified quadratic:
$$(x+3)(x-2) = 0$$
Solve for $x$:
$x+3=0 \implies x=-3$; $x-2=0 \implies x=2$

Step4: Find y for x=2

Substitute $x=2$ into $y=-2x+\frac{21}{4}$:
$$y = -2(2) + \frac{21}{4} = -4 + \frac{21}{4} = \frac{-16+21}{4} = \frac{5}{4}$$

Step5: Find y for x=-3

Substitute $x=-3$ into $y=-2x+\frac{21}{4}$:
$$y = -2(-3) + \frac{21}{4} = 6 + \frac{21}{4} = \frac{24+21}{4} = \frac{45}{4}$$

Answer:

$(2, \frac{5}{4})$, $(-3, \frac{45}{4})$