Question

determine the number of solutions for each system of equations. justify your answers.
a. \\(y = 3x + 7\\) and \\(y = 3x - 7\\)
b. \\(x - 2y = -5\\) and \\(4x - 8y = -20\\)
c. \\(y = -6x + 1\\) and \\(y = 6x + 1\\)

Explanation:

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<reasoning>

Analyze system (a)

\[

$$\begin{aligned} &y = 3x + 7 \quad (m_1 = 3, \, b_1 = 7) \\ &y = 3x - 7 \quad (m_2 = 3, \, b_2 = -7) \\ &m_1 = m_2 \text{ and } b_1 eq b_2 \implies 0 \text{ solutions (parallel lines)} \end{aligned}$$

\]

Analyze system (b)

\[

$$\begin{aligned} &x - 2y = -5 \implies 4(x - 2y) = 4(-5) \implies 4x - 8y = -20 \\ &\text{Both equations are identical} \implies \infty \text{ solutions (coincident lines)} \end{aligned}$$

\]

Analyze system (c)

\[

$$\begin{aligned} &y = -6x + 1 \quad (m_1 = -6, \, b_1 = 1) \\ &y = 6x + 1 \quad (m_2 = 6, \, b_2 = 1) \\ &m_1 eq m_2 \implies 1 \text{ solution (intersecting lines)} \end{aligned}$$

\]
</reasoning>

<answer>
a. No solution (0 solutions)
b. Infinitely many solutions
c. One solution (1 solution)
</answer>

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"question_type": "Multi-part",
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"Number of Solutions"
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Answer:

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"new_concepts": [],
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<reasoning>

Analyze system (a)

\[

$$\begin{aligned} &y = 3x + 7 \quad (m_1 = 3, \, b_1 = 7) \\ &y = 3x - 7 \quad (m_2 = 3, \, b_2 = -7) \\ &m_1 = m_2 \text{ and } b_1 eq b_2 \implies 0 \text{ solutions (parallel lines)} \end{aligned}$$

\]

Analyze system (b)

\[

$$\begin{aligned} &x - 2y = -5 \implies 4(x - 2y) = 4(-5) \implies 4x - 8y = -20 \\ &\text{Both equations are identical} \implies \infty \text{ solutions (coincident lines)} \end{aligned}$$

\]

Analyze system (c)

\[

$$\begin{aligned} &y = -6x + 1 \quad (m_1 = -6, \, b_1 = 1) \\ &y = 6x + 1 \quad (m_2 = 6, \, b_2 = 1) \\ &m_1 eq m_2 \implies 1 \text{ solution (intersecting lines)} \end{aligned}$$

\]
</reasoning>

<answer>
a. No solution (0 solutions)
b. Infinitely many solutions
c. One solution (1 solution)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Number of Solutions"
]
}
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