which of these statements is correct?

- the system of linear equations $6x - 5y = 8$ and $12x - 10y = 16$ has no solution.
- the system of linear equations $7x + 2y = 6$ and $14x + 4y = 16$ has an infinite number of solutions.
- the system of linear equations $8x - 3y = 10$ and $16x - 6y = 22$ has no solution.
- the system of linear equations $9x + 6y = 14$ and $18x + 12y = 26$ has an infinite number of solutions.

Question

which of these statements is correct?

- the system of linear equations $6x - 5y = 8$ and $12x - 10y = 16$ has no solution.
- the system of linear equations $7x + 2y = 6$ and $14x + 4y = 16$ has an infinite number of solutions.
- the system of linear equations $8x - 3y = 10$ and $16x - 6y = 22$ has no solution.
- the system of linear equations $9x + 6y = 14$ and $18x + 12y = 26$ has an infinite number of solutions.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "System of Equations", "Infinite Solutions Condition" ], "new_concepts": [ "No Solution Condition" ], "current_concepts": [ "System of Equations", "Infinite Solutions Condition", "No Solution Condition" ] } ## Analyze the first system Using the *System of Equations* and *Infinite Solutions Condition* knowledge points $$ \begin{aligned} &\text{System: } 6x - 5y = 8 \text{ and } 12x - 10y = 16\ &\text{Multiply first equation by } 2: 12x - 10y = 16\ &\text{Both equations are identical, so there are infinitely many solutions.} \end{aligned} $$ ## Analyze the second system Using the *System of Equations* and *Infinite Solutions Condition* knowledge points $$ \begin{aligned} &\text{System: } 7x + 2y = 6 \text{ and } 14x + 4y = 16\ &\text{Multiply first equation by } 2: 14x + 4y = 12\ &\text{Since } 14x + 4y = 12 \text{ and } 14x + 4y = 16 \text{ cannot both be true, there is no solution.} \end{aligned} $$ ## Analyze the third system Using the *System of Equations* knowledge point $$ \begin{aligned} &\text{System: } 8x - 3y = 10 \text{ and } 16x - 6y = 22\ &\text{Multiply first equation by } 2: 16x - 6y = 20\ &\text{Since } 16x - 6y = 20 \text{ and } 16x - 6y = 22 \text{ cannot both be true, there is no solution.} \end{aligned} $$ ## Analyze the fourth system Using the *System of Equations* and *Infinite Solutions Condition* knowledge points $$ \begin{aligned} &\text{System: } 9x + 6y = 14 \text{ and } 18x + 12y = 26\ &\text{Multiply first equation by } 2: 18x + 12y = 28\ &\text{Since } 18x + 12y = 28 \text{ and } 18x + 12y = 26 \text{ cannot both be true, there is no solution.} \end{aligned} $$

(A) The system of linear equations $6x-5y=8$ and $12x-10y=16$ has no solution.

(B) The system of linear equations $7x+2y=6$ and $14x+4y=16$ has an infinite number of solutions.

(C) The system of linear equations $8x-3y=10$ and $16x-6y=22$ has no solution.

(D) The system of linear equations $9x+6y=14$ and $18x+12y=26$ has an infinite number of solutions.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "System of Equations" ] }

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Question

Explanation:

Analyze the first system

Analyze the second system

Analyze the third system

Analyze the fourth system

Answer: