what are the solutions of the following system?

$\begin{cases} x^2 + y^2 = 25 \\ 2x + y = -5 \end{cases}$

(0, -5) and (-5, 5)
(0, -5) and (5, -15)
(0, -5) and (-4, 3)
(0, -5) and (4, -13)

Question

what are the solutions of the following system?

$\begin{cases} x^2 + y^2 = 25 \\ 2x + y = -5 \end{cases}$

(0, -5) and (-5, 5)
(0, -5) and (5, -15)
(0, -5) and (-4, 3)
(0, -5) and (4, -13)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Systems of Equations", "System of Quadratic and Linear Equations" ], "new_concepts": [], "current_concepts": [ "Systems of Equations", "System of Quadratic and Linear Equations" ] } ## Express y in terms of x from the linear equation $$ y = -2x - 5 $$ ## Substitute into the quadratic equation and solve for x $$ \begin{aligned} x^2 + (-2x - 5)^2 &= 25 \ x^2 + (4x^2 + 20x + 25) &= 25 \ 5x^2 + 20x &= 0 \ 5x(x + 4) &= 0 \ x = 0 \quad &\text{or} \quad x = -4 \end{aligned} $$ ## Find the corresponding y-values $$ \begin{aligned} \text{For } x = 0: \quad y &= -2(0) - 5 = -5 \implies (0, -5) \ \text{For } x = -4: \quad y &= -2(-4) - 5 = 3 \implies (-4, 3) \end{aligned} $$

(A) $(0, -5)$ and $(-5, 5)$

(B) $(0, -5)$ and $(5, -15)$

(C) $(0, -5)$ and $(-4, 3)$

(D) $(0, -5)$ and $(4, -13)$

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Question

Explanation:

Express y in terms of x from the linear equation

Substitute into the quadratic equation and solve for x

Find the corresponding y-values

Answer: