Question

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

Explanation:

Step1: Set equations equal

Since both equal \( y \), set \( -2x + \frac{21}{4} = x^2 - x - \frac{3}{4} \).

Step2: Simplify equation

Multiply all terms by 4: \( -8x + 21 = 4x^2 - 4x - 3 \).
Rearrange: \( 4x^2 + 4x - 24 = 0 \) → Divide by 4: \( x^2 + x - 6 = 0 \).

Step3: Solve quadratic

Factor: \( (x + 3)(x - 2) = 0 \) → \( x = -3 \) or \( x = 2 \).

Step4: Find \( y \) for \( x = -3 \)

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

Step5: Find \( y \) for \( x = 2 \)

Substitute \( x = 2 \) into \( y = -2x + \frac{21}{4} \):
\( y = -4 + \frac{21}{4} = \frac{-16 + 21}{4} = \frac{5}{4} \).
So point: \( (2, \frac{5}{4}) \).

Answer:

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