Find the slope of the line that passes through the points $(2,-5)$ and $(7,1)$.
Step 1: Choose $\left(x_1, y_1\right)$.
$x_1=$ $\square$ , $y_1=$ $\square$
Step 2: Identify $\left(x_2, y_2\right)$.
$x_2=$ $\square$ , $y_2=$ $\square$
Answer (1)
Identify the coordinates of the two points: ( x 1 , y 1 ) = ( 2 , − 5 ) and ( x 2 , y 2 ) = ( 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 ) .
Calculate the slope: m = 5 6 .
The slope of the line is 5 6 .
Explanation
Understanding the Problem We are given two points, ( 2 , − 5 ) and ( 7 , 1 ) , and we need to find the slope of the line that passes through these points. 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
Identifying the Coordinates Let's identify the coordinates of the two points. We can choose ( x 1 , y 1 ) = ( 2 , − 5 ) and ( x 2 , y 2 ) = ( 7 , 1 ) . So, we have: x 1 = 2 y 1 = − 5 x 2 = 7 $y_2 = 1
Calculating the Slope Now, we substitute these values into the slope formula: m = 7 − 2 1 − ( − 5 ) = 5 1 + 5 = 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 or ramps, engineers use the concept of slope to ensure they are not too steep for vehicles or people to navigate safely. In economics, the slope of a supply or demand curve can indicate how responsive the quantity supplied or demanded is to changes in price. Moreover, in physics, the slope of a velocity-time graph represents acceleration. These examples highlight how the mathematical concept of slope is fundamental in various fields, enabling us to analyze and design systems effectively.
