When the two equations are graphed on the coordinate plane, how are the graphs similar?
Understand the Problem
The question is asking to analyze and compare the graphs of two linear equations representing the costs for existing and new customers of a cable company. It requires understanding the similarities in their graphing on a coordinate plane, specifically focusing on aspects like slope and y-intercept.
Answer
The graphs are linear and parallel, both having a slope of 70, but with different y-intercepts (0 for existing customers and 100 for new customers).
Answer for screen readers
The graphs are similar in that both are linear with the same slope of 70, indicating they grow at the same rate but differ in their starting points, with the new customer line having a y-intercept of 100 compared to 0 for existing customers.
Steps to Solve
- Identify the equations The two equations given are:
- Existing Customer: ( y = 70x )
- New Customer: ( y = 70x + 100 )
- Determine the slopes The slope of a linear equation ( y = mx + b ) is represented by ( m ).
- For Existing Customers, ( m = 70 ).
- For New Customers, ( m = 70 ).
Thus, both lines have the same slope of 70.
- Determine the y-intercepts The y-intercept ( b ) is the value of ( y ) when ( x = 0 ).
- For Existing Customers, ( b = 0 ).
- For New Customers, ( b = 100 ).
Therefore, the y-intercept for the line representing new customers is 100.
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Graphing the equations When graphed, both lines will be parallel since they have the same slope. However, the line for the new customers will be shifted upwards because of its higher y-intercept.
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Conclusion about similarities Both graphs are similar in that they are linear and parallel. They share the same rate of change (slope) but differ in their starting point (y-intercept).
The graphs are similar in that both are linear with the same slope of 70, indicating they grow at the same rate but differ in their starting points, with the new customer line having a y-intercept of 100 compared to 0 for existing customers.
More Information
The linearity and parallel nature imply that both pricing strategies increase costs at the same rate, but the additional fee for new customers modifies their overall cost starting point. This is a common structure in pricing models that distinguish between different customer groups.
Tips
- Confusing slope with y-intercept: Make sure to clearly identify both elements in a linear equation.
- Not recognizing parallel lines: Parallel lines have the same slope but different y-intercepts, which is essential in comparing their graphs.
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