What's the slope of a line that passes through $(-12,15)$ and $(0,4)$?
A) $\frac{12}{11}$
B) $\frac{-12}{11}$
C) $\frac{-11}{12}$
D) $\frac{11}{12}$
Answer (1)
The problem provides two points, ( − 12 , 15 ) and ( 0 , 4 ) , and asks for the slope of the line passing through them.
The slope formula is m = x 2 − x 1 y 2 − y 1 .
Substituting the given points into the formula gives m = 0 − ( − 12 ) 4 − 15 = 12 − 11 .
The slope of the line is 12 − 11 .
Explanation
Understanding the Problem We are given two points through which a line passes: ( − 12 , 15 ) and ( 0 , 4 ) . Our goal is to find the slope of this line. The slope, often denoted as m , represents the rate of change of the line, or how much the y -value changes for every unit change in the x -value.
Recalling the Slope Formula The formula to calculate the slope ( m ) of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by:
m = x 2 − x 1 y 2 − y 1
This formula calculates the change in y divided by the change in x , which gives us the slope.
Substituting the Values Let's assign the given points to the variables in the formula:
( x 1 , y 1 ) = ( − 12 , 15 ) ( x 2 , y 2 ) = ( 0 , 4 )
Now, substitute these values into the slope formula:
m = 0 − ( − 12 ) 4 − 15
Calculating the Slope Now, let's simplify the expression to find the slope:
m = 12 − 11
So, the slope of the line is − 12 11 .
Final Answer The slope of the line that passes through the points ( − 12 , 15 ) and ( 0 , 4 ) is − 12 11 . Therefore, the correct answer is C) 12 − 11 .
Examples
Understanding the slope of a line is crucial in various 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 (larger positive value) requires more effort, while a gentler slope (smaller positive value) makes it easier. Similarly, in economics, the slope of a supply or demand curve indicates how responsive the quantity supplied or demanded is to changes in price. A steeper slope suggests a greater responsiveness.
