Find the slope of the line that passes through the points shown in the table.
| x | y |
| --- | --- |
| -14 | 8 |
| -7 | 6 |
| 0 | 4 |
| 7 | 2 |
| 14 | 0 |
The slope of the line that passes through the points in the table is $\square$
Answer (1)
Use the slope formula m = x 2 − x 1 y 2 − y 1 .
Choose two points from the table, for example, ( − 14 , 8 ) and ( − 7 , 6 ) .
Calculate the slope: m = − 7 − ( − 14 ) 6 − 8 = − 7 2 .
The slope of the line is − 7 2 .
Explanation
Understanding the Problem We are given a table of x and y values that represent points on a line. Our goal is to find the slope of this line. The slope of a line is a measure of its steepness and direction. It tells us how much the y -value changes for every unit change in the x -value.
The Slope Formula To find the slope of a line, we can use the slope formula, which is: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are any two points on the line.
Calculating the Slope Let's choose the first two points from the table: ( − 14 , 8 ) and ( − 7 , 6 ) . Plugging these values into the slope formula, we get: m = − 7 − ( − 14 ) 6 − 8 = − 7 + 14 − 2 = 7 − 2 = − 7 2 So, the slope of the line is − 7 2 .
Verifying the Slope To make sure we didn't make a mistake, let's choose two different points and calculate the slope again. Let's use the points ( 0 , 4 ) and ( 14 , 0 ) . Plugging these values into the slope formula, we get: m = 14 − 0 0 − 4 = 14 − 4 = − 7 2 The slope is still − 7 2 , which confirms our previous calculation.
Final Answer Therefore, the slope of the line that passes through the points in the table is − 7 2 .
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 easy or difficult it is to use. A steeper slope (larger absolute value) requires more effort to ascend, while a gentler slope (smaller absolute value) is easier. Similarly, in economics, the slope of a supply or demand curve can tell us how sensitive the quantity supplied or demanded is to changes in price. In construction, the slope of a roof is important for water runoff and structural integrity. The concept of slope is also used in computer graphics, physics, and many other fields.
