what is one solution of the following system?

$\begin{cases} 2y - 2x = 12 \\ x^2 + y^2 = 36 \end{cases}$

$o$ $(-6, 0)$
$o$ $(-2, 4)$
$o$ $(0, -6)$
$o$ $(4, -2)$

Question

what is one solution of the following system?

$\begin{cases} 2y - 2x = 12 \\ x^2 + y^2 = 36 \end{cases}$

$o$ $(-6, 0)$
$o$ $(-2, 4)$
$o$ $(0, -6)$
$o$ $(4, -2)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Systems of Equations from Single Equations" ], "new_concepts": [ "Nonlinear Systems of Equations", "Circle and Line Intersections" ], "current_concepts": [ "Systems of Equations from Single Equations", "Nonlinear Systems of Equations", "Circle and Line Intersections" ] } ## Express y in terms of x Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} 2y - 2x &= 12 \ 2y &= 2x + 12 \ y &= x + 6 \end{aligned} $$ ## Substitute into the second equation Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} x^2 + y^2 &= 36 \ x^2 + (x + 6)^2 &= 36 \end{aligned} $$ ## Solve the quadratic equation Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} x^2 + x^2 + 12x + 36 &= 36 \ 2x^2 + 12x &= 0 \ 2x(x + 6) &= 0 \ x = 0 \quad &\text{or} \quad x = -6 \end{aligned} $$ ## Determine the corresponding y-values Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} \text{If } x = 0: \quad y &= 0 + 6 = 6 \implies (0, 6) \ \text{If } x = -6: \quad y &= -6 + 6 = 0 \implies (-6, 0) \end{aligned} $$ ## Match with the given options Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} \text{Solutions: } &(-6, 0) \text{ and } (0, 6) \ \text{Matching Option: } &(-6, 0) \end{aligned} $$

(A) $(-6, 0)$

(B) $(-2, 4)$

(C) $(0, -6)$

(D) $(4, -2)$

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QUESTION IMAGE

Question

Explanation:

Express y in terms of x

Substitute into the second equation

Solve the quadratic equation

Determine the corresponding y-values

Match with the given options

Answer: