Question

1. 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

\[

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

\]

Analyze system b

\[

$$\begin{aligned} &x - 2y = -5 \implies 4x - 8y = -20 \\ &4x - 8y = -20 \\ &\text{Identical lines} \implies \infty \text{ solutions} \end{aligned}$$

\]

Analyze system c

\[

$$\begin{aligned} &y = -6x + 1 \\ &y = 6x + 1 \\ &m_1 = -6 eq m_2 = 6 \\ &\text{Intersecting lines} \implies 1 \text{ solution} \end{aligned}$$

\]
</reasoning>

<answer>

Question 6a

0 solutions

Question 6b

Infinite solutions

Question 6c

1 solution
</answer>

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"subject": "Mathematics",
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"Mathematics",
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"System of Equations"
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Answer:

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"Number of Solutions"
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<reasoning>

Analyze system a

\[

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

\]

Analyze system b

\[

$$\begin{aligned} &x - 2y = -5 \implies 4x - 8y = -20 \\ &4x - 8y = -20 \\ &\text{Identical lines} \implies \infty \text{ solutions} \end{aligned}$$

\]

Analyze system c

\[

$$\begin{aligned} &y = -6x + 1 \\ &y = 6x + 1 \\ &m_1 = -6 eq m_2 = 6 \\ &\text{Intersecting lines} \implies 1 \text{ solution} \end{aligned}$$

\]
</reasoning>

<answer>

Question 6a

0 solutions

Question 6b

Infinite solutions

Question 6c

1 solution
</answer>

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