Find the slope of the line containing the points $(-3,-6)$ and $(4,7)$.
A. 1
B. $\frac{13}{7}$
C. $\frac{1}{13}$
D. $\frac{7}{13}$
Answer (2)
Identify the coordinates of the two given points: ( − 3 , − 6 ) and ( 4 , 7 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 4 − ( − 3 ) 7 − ( − 6 ) .
Simplify to find the slope: m = 7 13 .
7 13
Explanation
Understanding the Problem We are given two points on a line, ( − 3 , − 6 ) and ( 4 , 7 ) , and we need to find the slope of the line that passes through these points.
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.
Identifying Coordinates Let's identify the coordinates: x 1 = − 3 , y 1 = − 6 x 2 = 4 , y 2 = 7
Substituting Values Now, substitute these values into the slope formula: m = 4 − ( − 3 ) 7 − ( − 6 )
Simplifying the Expression Simplify the expression: m = 4 + 3 7 + 6 = 7 13
Final Answer Therefore, the slope of the line containing the points ( − 3 , − 6 ) and ( 4 , 7 ) is 7 13 .
Examples
Understanding the slope is crucial in many real-world applications. For example, if you're analyzing the steepness of a hill for a hiking trail, the slope helps determine the difficulty. A larger slope means a steeper climb. Similarly, in construction, the slope of a roof is essential for water runoff. Knowing how to calculate slope from two points allows engineers and builders to design structures that function effectively and safely. This concept extends to economics, where understanding the slope of a cost function can help businesses optimize production.
The slope of the line passing through the points ( − 3 , − 6 ) and ( 4 , 7 ) is 7 13 .
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