Find the slope of the line passing through the points given in the table.
| x | y |
| --- | ----- |
| 4 | 3 |
| 9 | -9.5 |
| 14 | -22 |
| 19 | -34.5 |
Answer (1)
Identify two points from the table: ( 4 , 3 ) and ( 9 , − 9.5 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates: m = 9 − 4 − 9.5 − 3 .
Calculate the slope: m = − 2.5 . The slope of the line is − 2.5 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the slope of the line passing through these points. The slope of a line between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 We can choose any two points from the table to calculate the slope.
Calculating the Slope Let's choose the first two points from the table: ( 4 , 3 ) and ( 9 , − 9.5 ) . Plugging these values into the slope formula, we get: m = 9 − 4 − 9.5 − 3 = 5 − 12.5 = − 2.5 So, the slope is − 2.5 .
Verifying the Slope To verify our result, let's choose another pair of points, say ( 14 , − 22 ) and ( 19 , − 34.5 ) . Using the slope formula again: m = 19 − 14 − 34.5 − ( − 22 ) = 5 − 34.5 + 22 = 5 − 12.5 = − 2.5 Since we obtain the same slope using a different pair of points, we can be confident in our answer.
Final Answer Therefore, the slope of the line passing through the points given in the table is − 2.5 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, consider a ramp for wheelchair access. The slope of the ramp determines how steep it is. A steeper slope requires more effort to ascend, while a gentler slope is easier. Building codes often specify maximum allowable slopes for ramps to ensure accessibility. Similarly, in road construction, the slope (or grade) of a road affects the ease with which vehicles can travel up or down it. Civil engineers carefully calculate and manage slopes to optimize safety and efficiency.
