Find the slope of the line that passes through the points $(2,-5)$ and $(7,1)$.
Answer (2)
Identify the coordinates of the two points: ( 2 , − 5 ) and ( 7 , 1 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 7 − 2 1 − ( − 5 ) .
Simplify to find the slope: m = 5 6 .
5 6
Explanation
Understanding the Problem We are given two points, ( 2 , − 5 ) and ( 7 , 1 ) , and we want to find the slope of the line that passes through these points.
Recalling the Slope Formula The slope of a line passing through 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 where m represents the slope.
Identifying the Coordinates Let's identify the coordinates of our points: ( x 1 , y 1 ) = ( 2 , − 5 ) and ( x 2 , y 2 ) = ( 7 , 1 ) .
Substituting and Simplifying Now, we substitute these values into the slope formula: m = 7 − 2 1 − ( − 5 ) m = 7 − 2 1 + 5 m = 5 6
Final Answer Therefore, the slope of the line that passes through the points ( 2 , − 5 ) and ( 7 , 1 ) is 5 6 .
Examples
Understanding slope is crucial in many real-world applications. For instance, when designing roads, engineers use the concept of slope to determine the steepness of the road. A steeper slope means a more difficult climb for vehicles. Similarly, in construction, the slope of a roof is essential for proper water runoff. A well-calculated slope prevents water from pooling and causing damage. In finance, the slope of a stock's price trend can indicate its rate of growth or decline, helping investors make informed decisions. These examples highlight how the mathematical concept of slope is fundamental in various practical scenarios.
The slope of the line that passes through the points (2, -5) and (7, 1) is calculated using the slope formula, resulting in a slope of 5 6 . This shows how much y changes for a change in x between the two points. Thus, the slope indicates a rise of 6 units for every 5 units of run.
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