Where Lines with Different Slopes Meet On the same coordinate axes, draw one line with vertical intercept 2 and slope 3 and another with vertical intercept 4 and slope 1. Do these lines cross? If so, do they cross to the right or left of the vertical axis? In general, if one line has its vertical intercept below the vertical intercept of another, what conditions on the slope will ensure that the lines cross to the right of the vertical axis? Give me a accurate graph

The two lines \( y = 3x + 2 \) and \( y = x + 4 \) do cross, and their point of intersection is at \( (1.0, 5.0) \). This intersection occurs to the right of the vertical (y) axis, as \( x = 1.0 \).

### General Case:
If one line has its vertical intercept below another, the lines will cross to the right of the vertical axis if the line with the steeper slope (greater value) also has the lower vertical intercept. In this case, since \( y = 3x + 2 \) has both a steeper slope (3) and a lower intercept (2) compared to \( y = x + 4 \), they cross to the right of the y-axis.

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Here are five related questions for further exploration:
1. How can we generalize the point of intersection for two linear equations?
2. What happens if the two lines are parallel? How can we detect that algebraically?
3. How would the intersection change if the slopes were negative?
4. How do vertical and horizontal lines interact with other lines in terms of intersection?
5. How does the intersection point move if the slopes are fixed but intercepts are changed?

**Tip:** The point where two lines cross can be found by solving their equations simultaneously, either by substitution or elimination.

Where Lines with Different Slopes Meet On the same coordinate axes, draw one line with vertical intercept 2 and slope 3 and another with vertical intercept 4 and slope 1. Do these lines cross? If so, do they cross to the right or left of the vertical axis? In general, if one line has its vertical intercept below the vertical intercept of another, what conditions on the slope will ensure that the lines cross to the right of the vertical axis?

Explore how two lines with different slopes and intercepts intersect. This example draws two lines with slopes 3 and 1, and vertical intercepts 2 and 4, respectively. Understand the conditions that ensure the lines cross to the right of the vertical axis. Ideal for students in Grades 9-11.

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