Which gives the slope of the line containing the points (2, 7) and (3, 4)?

Understand the Problem

The question is asking to determine the correct formula for calculating the slope of the line that passes through two given points (2, 7) and (3, 4). We need to analyze the provided options for the slope equation.

Answer

The correct answer is option (c): $$ m = \frac{7 - 4}{2 - 3} $$

Steps to Solve

Understand the Slope Formula

The slope ($m$) of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Identify the Points

In this problem, the points given are ((2, 7)) and ((3, 4)). Here, ((x_1, y_1) = (2, 7)) and ((x_2, y_2) = (3, 4)).

Substitute Values into the Slope Formula

Plugging in the coordinates into the slope formula:
$$ m = \frac{4 - 7}{3 - 2} $$

Calculate the Numerator and Denominator

Calculate the difference in the (y)-coordinates and (x)-coordinates:
Numerator: (4 - 7 = -3)
Denominator: (3 - 2 = 1)

Calculate the Slope

Thus, the slope is:
$$ m = \frac{-3}{1} = -3 $$

Analyze the Options

Now we check which option correctly represents this slope calculation.

Option (a): (m = \frac{2 - 7}{3 - 4}) simplifies to (m = \frac{-5}{-1} = 5) (not correct).
Option (b): (m = \frac{2}{7}) (not correct).
Option (c): (m = \frac{7 - 4}{2 - 3}) simplifies to (m = \frac{3}{-1} = -3) (correct).

The correct answer is option (c):
$$ m = \frac{7 - 4}{2 - 3} $$

More Information

The slope of a line indicates its steepness and direction. A negative slope means the line descends from left to right, while a positive slope indicates it ascends. The slope calculated here indicates that for each unit increase in (x), (y) decreases by 3 units.

Tips

Confusing the order of coordinates; make sure to identify ((x_1, y_1)) and ((x_2, y_2)) correctly.
Incorrectly calculating the differences in the coordinates, leading to wrong slope values.

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