When two lines on a graph have the same slope but different y-intercepts, it means that they are parallel to each other but do not meet (intersect) at any point. Here's a clearer explanation:
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Understanding Slope and Y-Intercept:
- The slope of a line measures how steep it is and the direction in which it rises or falls. It is often represented by the letter \( m \).
- The y-intercept is the point where the line crosses the y-axis, represented by the letter \( b \) in the equation of a line in slope-intercept form: \( y = mx + b \).
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Same Slope:
- If two lines have the same slope (\( m_1 = m_2 \)), it indicates that they rise or fall at the same rate. For example, a slope of 2 means that for every unit you move to the right (in the positive x direction), the line moves up 2 units (in the positive y direction). This similarity in slope means that they are parallel.
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Different Y-Intercepts:
- Different y-intercepts (\( b_1 \neq b_2 \)) imply that the lines start at different points on the y-axis. For instance, one line might intersect the y-axis at (0, 3), while the other intersects it at (0, 1). This means that even though both lines are moving in the same direction and at the same rate, they are vertically offset from each other.
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Visual Representation:
- In a graph, you will see two lines that never cross each other — one will always be above or below the other but still maintaining the same parallel direction.
Conclusion
In summary, two lines with the same slope represent parallel lines that do not intersect, while their different y-intercepts indicate that they start from different heights on the y-axis. This distinction is fundamental in linear equations and helps to visualize and understand linear relationships in algebra, geometry, and other areas of mathematics.