The slope for the line with points $(-1,5)$ and $(2,2)$ is $\frac{3}{3}$. True or False?
Answer (2)
Recall the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the given points ( − 1 , 5 ) and ( 2 , 2 ) into the formula: m = 2 − ( − 1 ) 2 − 5 .
Simplify the expression to find the slope: m = 3 − 3 = − 1 .
Compare the calculated slope with the given slope 3 3 and conclude that the statement is false: F a l se .
Explanation
Analyze the problem and recall the slope formula. We are given two points on a line, ( − 1 , 5 ) and ( 2 , 2 ) , and we want to determine if the slope of the line is 3 3 . 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
In this case, ( x 1 , y 1 ) = ( − 1 , 5 ) and ( x 2 , y 2 ) = ( 2 , 2 ) .
Calculate the slope. Substitute the given coordinates into the slope formula:
m = 2 − ( − 1 ) 2 − 5
Simplify the expression:
m = 2 + 1 − 3 = 3 − 3 = − 1
The calculated slope is − 1 .
Compare the calculated slope with the given slope and conclude. The problem states that the slope is 3 3 , which simplifies to 1 . However, our calculated slope is − 1 . Since 1 = − 1 , the given slope is incorrect.
Therefore, the statement 'The slope for the line with points ( − 1 , 5 ) and ( 2 , 2 ) is 3 3 ' is false.
Examples
Understanding slope is crucial in many real-world applications. For example, 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 using a wheelchair on a ramp. By accurately calculating and controlling the slope, engineers can create designs that meet specific safety and usability standards.
The correct slope between the points ( − 1 , 5 ) and ( 2 , 2 ) is − 1 , not 1 (which is what 3 3 simplifies to). Therefore, the statement is false.
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