Find the slope of the line that passes through these two points:
[tex]$(-3,-2) ;(2,1)$[/tex]
Simplify your answer completely.
Slope = [?]
Slope Formula: [tex]$\frac{y_2-y_1}{x_2-x_1}$[/tex]
Answer (1)
Identify the coordinates of the two points: ( − 3 , − 2 ) and ( 2 , 1 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 2 − ( − 3 ) 1 − ( − 2 ) .
Simplify to find the slope: m = 5 3 .
The slope of the line is 5 3 .
Explanation
Understanding the Problem We are given two points, ( − 3 , − 2 ) and ( 2 , 1 ) , and we want to find the slope of the line that passes through them. We are also given the slope formula:
Stating the Slope Formula The slope formula is given by: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points.
Substituting the Values Let's assign the coordinates: ( x 1 , y 1 ) = ( − 3 , − 2 ) and ( x 2 , y 2 ) = ( 2 , 1 ) .
Now, substitute these values into the slope formula: m = 2 − ( − 3 ) 1 − ( − 2 )
Simplifying the Expression Simplify the expression: m = 2 + 3 1 + 2 = 5 3 So, the slope of the line is 5 3 .
Final Answer Therefore, the slope of the line that passes through the points ( − 3 , − 2 ) and ( 2 , 1 ) is 5 3 .
Examples
Understanding slope is crucial in many real-world applications. For instance, when designing roads or ramps, engineers need to calculate the slope to ensure they are safe and accessible. A steeper slope requires more effort to climb, whether it's a car driving up a hill or a person walking up a ramp. By calculating the slope, engineers can optimize the design for efficiency and safety. Similarly, in construction, the slope of a roof is essential for proper water runoff and preventing leaks. Knowing how to calculate slope helps in various fields, ensuring designs are practical and functional.
