select the correct answer.

what is the solution set of this system of equations?

$\begin{cases} y = -x^2 + 2x + 4 \\ x + y = 4 \end{cases}$

(0, 4) and (4, 0)
(7, -3) and (0, 4)
(0, 4) and (3, 1)
(7, -3) and (4, 0)

Question

select the correct answer.

what is the solution set of this system of equations?

$\begin{cases} y = -x^2 + 2x + 4 \\ x + y = 4 \end{cases}$

(0, 4) and (4, 0)
(7, -3) and (0, 4)
(0, 4) and (3, 1)
(7, -3) and (4, 0)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Linear-Quadratic Systems", "Algebraic Rearrangement", "Quadratic Formula Application" ], "new_concepts": [], "current_concepts": [ "Linear-Quadratic Systems", "Algebraic Rearrangement", "Quadratic Formula Application" ] } ## Substitute the linear equation into the quadratic equation $$ \begin{aligned} &x + y = 4 \implies y = 4 - x\ &4 - x = -x^2 + 2x + 4 \end{aligned} $$ ## Solve the resulting quadratic equation for x $$ \begin{aligned} &x^2 - 3x = 0\ &x(x - 3) = 0\ &x = 0 \quad \text{or} \quad x = 3 \end{aligned} $$ ## Find the corresponding y-values $$ \begin{aligned} &\text{For } x = 0: y = 4 - 0 = 4 \implies (0, 4)\ &\text{For } x = 3: y = 4 - 3 = 1 \implies (3, 1) \end{aligned} $$

(0, 4) and (4, 0)

(7, -3) and (0, 4)

(0, 4) and (3, 1)

(7, -3) and (4, 0)

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Linear-Quadratic Systems" ] }

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QUESTION IMAGE

Question

Explanation:

Substitute the linear equation into the quadratic equation

Solve the resulting quadratic equation for x

Find the corresponding y-values

Answer: