Question

the graph shows the following system:

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$$\begin{cases} y = \\frac{1}{2}x - 2 \\\\ y = \\frac{1}{2}x + 4 \\end{cases}$$

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q. what can you conclude about the system?

a. this system cannot be graphed.
b. the solution is not shown on the graph.

Explanation:

Identify the system of equations

Using the Systems of Equations knowledge point

$$
LATEXBLOCK0
$$

Analyze the slopes of the lines

Both equations are in slope-intercept form \(y = mx + b\).
The slope of the first line is \(m_1 = \frac{1}{2}\).
The slope of the second line is \(m_2 = \frac{1}{2}\).

Compare the y-intercepts

The y-intercept of the first line is \(b_1 = -2\).
The y-intercept of the second line is \(b_2 = 4\).

Determine geometric relationship

Since the slopes are equal (\(m_1 = m_2\)) and the y-intercepts are different (\(b_1
eq b_2\)), the two lines are parallel.
Parallel lines never intersect.

Conclude the system's solution

Because the lines never intersect, there is no common point \((x, y)\) that satisfies both equations.
Therefore, the system has no solution.

Answer:

• A. This system cannot be graphed.
• B. The solution is not shown on the graph.
• C. There is no solution. (Correct answer)