Let's go through each of these questions one by one.

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1. Which equation represents a line that intersects $y = 5x - 6$ at exactly one point?

The question is asking for a line that intersects the given line $y = 5x - 6$ at exactly one point, which means the two lines must not be parallel (i.e., they must have different slopes). The given line has a slope of 5. To find the correct equation, we need to look for a line with a different slope.

Let's analyze the options:

• A. $y = 3x + 4$: The slope here is 3, which is different from 5, so this line will intersect $y = 5x - 6$ at exactly one point.
• B. $y = 3x$: The slope here is also 3, which is different from 5, so this will also intersect at one point.
• C. $y = 2x - 6$: The slope here is 2, which is different from 5, so this will intersect at one point.
• D. $y = 5x - 6$: This has the same slope as the original equation and is exactly the same line, so they will not intersect at exactly one point; they are coincident.

Thus, the answer is A, B, or C.

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2. Which equation represents a line that is parallel but not coincident to $y = 5x - 6$?

For two lines to be parallel, they must have the same slope but different y-intercepts. The slope of $y = 5x - 6$ is 5, so we are looking for a line with the same slope but a different y-intercept.

Let's analyze the options:

• A. $y = 7x + 2$: The slope is 7, so this line is not parallel.
• B. $y = -x + 2$: The slope is -1, so this line is not parallel.
• C. $y = 5x + 1$: The slope is 5, which matches the original line, and the y-intercept is different. This line is parallel but not coincident.
• D. $y = -4x + 1$: The slope is -4, so this line is not parallel.

Thus, the answer is C.

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3. Which is true of the graphs of these two equations?

$p = 2x + 1$ and $q = x + 2$

We need to analyze whether the lines are the same, parallel, or if they intersect. The slopes and y-intercepts of the two equations are:

• For $p = 2x + 1$, the slope is 2.
• For $q = x + 2$, the slope is 1.

Since the slopes are different, the lines will intersect at exactly one point.

Let's analyze the options:

• A. The lines lie on top of one another: This would only be true if the two lines had identical slopes and y-intercepts, which they don't. So, this is false.
• B. The lines intersect at exactly one point: Since the slopes are different, the lines will indeed intersect at one point, so this is true.
• C. The lines have different slopes: This is also true, since the slopes of the lines are 2 and 1.
• D. The lines lie in the same plane, but they never intersect: This would only be true for parallel lines, but these lines are not parallel.

Thus, the correct answer is B. The lines intersect at exactly one point.

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Summary of Answers:

1. A, B, or C (Any of these can represent a line that intersects $y = 5x - 6$ at exactly one point).
2. C (The line that is parallel but not coincident to $y = 5x - 6$).
3. B (The lines intersect at exactly one point).

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Follow-up Questions:

1. How can you solve for the point where two lines intersect?
2. What happens to the intersection point if the slopes of the lines are equal?
3. How do you find the slope of a line given its equation in slope-intercept form?
4. What is the difference between parallel and coincident lines?
5. Can two lines with the same slope and different y-intercepts ever intersect?

Tip:

When analyzing the intersection of two lines, the key is to check their slopes. If they have different slopes, they will intersect at exactly one point. If the slopes are the same but the y-intercepts differ, they are parallel and never intersect.