20. The equation of line l is y = 3x/a+ 5, where a is a positive constant. If the value of a in this equation is doubled, then the resulting equation will represent a line whose slope is how many × the slope of line l?
Answer: C
The resulting equation will represent a line whose slope is 1/2 the slope of line l.
Doubling the value of a in the equation y = 3x/a + 5 changes the slope from 3/a to 3/(2a), which simplifies to (3/2)(1/a). This indicates that the new slope is 1/2 the original slope.
A) 2
This option is incorrect because if the value of a is doubled, the slope does not increase but rather decreases. The slope of the new line becomes 3/(2a), which is less than the original slope of 3/a, indicating that it cannot be twice the original slope.
B) 3/2
This option is also incorrect. It suggests that the new slope would be 3/2 times the original slope. However, since the new slope is actually 3/(2a), it is not greater than the original slope of 3/a but rather less.
C) 1/2
This option is correct. When the value of a is doubled, the slope of the equation changes from 3/a to 3/(2a), which simplifies to (3/2)(1/a). This indicates that the new slope is indeed 1/2 of the original slope.
D) 3/10
This option is incorrect because it suggests a specific numerical relationship that does not apply to the slope transformation in this context. The slope change is dependent on the factor of a, and the resulting slope does not equal 3/10 when a is doubled.
Conclusion
The correct answer, 1/2, reflects the precise mathematical relationship between the original slope and the slope after doubling a. All other options fail to accurately represent the relationship resulting from the change in a, confirming that the slope indeed decreases rather than increases or maintains a proportional relationship.