The graph of a line passes through the points $(3,2)$ and $(-4,1)$. Find the slope of this line.
Answer (1)
Identify the two points on the line: ( 3 , 2 ) and ( − 4 , 1 ) .
Recall the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = − 4 − 3 1 − 2 .
Simplify to find the slope: m = 7 1 .
Explanation
Understanding the Problem We are given two points, ( 3 , 2 ) and ( − 4 , 1 ) , that lie on a line. Our goal is to find the slope of this line.
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.
Substituting the Values Let's assign the given points to the variables in the formula. Let ( x 1 , y 1 ) = ( 3 , 2 ) and ( x 2 , y 2 ) = ( − 4 , 1 ) . Now, we substitute these values into the slope formula: m = − 4 − 3 1 − 2
Simplifying the Expression Now, we simplify the expression: m = − 7 − 1 = 7 1 So, the slope of the line is 7 1 .
Final Answer Therefore, the slope of the line that passes through the points ( 3 , 2 ) and ( − 4 , 1 ) is 7 1 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For instance, consider a ramp designed for wheelchair access. The slope of the ramp determines how easy or difficult it is to ascend. A steeper slope requires more effort, while a gentler slope is easier to navigate. Calculating the slope using two points on the ramp helps engineers ensure it meets accessibility standards, providing a safe and comfortable experience for users. This principle extends to various fields, including construction, transportation, and even financial analysis, where understanding rates of change is essential.
