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)
Recall the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the given points ( − 12 , 15 ) and ( 0 , 4 ) into the formula.
Calculate the slope: m = 0 − ( − 12 ) 4 − 15 = 12 − 11 .
The slope of the line is 12 − 11 .
Explanation
Understanding the Problem We are given two points, ( − 12 , 15 ) and ( 0 , 4 ) , and we need to find the slope of the line that passes through these points. The slope of a line is a measure of its steepness and direction. It tells us 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 given two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is: m = x 2 − x 1 y 2 − y 1 This formula represents the change in y divided by the change in x , often referred to as 'rise over run'.
Substituting the Values Let's assign the given points to the variables in the formula: x 1 = − 12 , y 1 = 15 x 2 = 0 , y 2 = 4 Now, substitute these values into the slope formula: m = 0 − ( − 12 ) 4 − 15
Calculating the Slope Now, let's simplify the expression: 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
Imagine you're hiking on a trail. The slope of the trail at any point tells you how steep the climb is. If the slope is positive, you're going uphill; if it's negative, you're going downhill. A larger slope (in absolute value) means a steeper trail. Understanding slope helps you anticipate how strenuous the hike will be. In construction, the slope of a roof is crucial for water runoff. A steeper slope allows water to drain more quickly, preventing leaks and damage. Similarly, civil engineers use slope calculations when designing roads and bridges to ensure safety and efficiency.
