What is the slope of the line with equation $y-3=-\frac{1}{2}(x-2)$?
Answer (2)
The equation is in point-slope form: y − y 1 = m ( x − x 1 ) .
Identify the coefficient of ( x − 2 ) in the given equation.
The coefficient of ( x − 2 ) is − 2 1 .
The slope of the line is − 2 1 .
Explanation
Understanding the Problem The equation of the line is given as y − 3 = − 2 1 ( x − 2 ) . We need to find the slope of this line. The equation is in point-slope form: y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Identifying the Slope Comparing the given equation y − 3 = − 2 1 ( x − 2 ) with the point-slope form y − y 1 = m ( x − x 1 ) , we can identify the slope m as the coefficient of ( x − 2 ) .
Determining the Slope In the equation y − 3 = − 2 1 ( x − 2 ) , the coefficient of ( x − 2 ) is − 2 1 . Therefore, the slope of the line is − 2 1 .
Final Answer The slope of the line with equation y − 3 = − 2 1 ( x − 2 ) is − 2 1 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For instance, consider the pitch of a roof, which is essentially the slope of the roofline. A steeper slope (larger absolute value) means a faster water runoff, which is important in areas with heavy rainfall. Similarly, in economics, the slope of a supply or demand curve indicates how sensitive the quantity supplied or demanded is to changes in price. The equation of a line is a fundamental concept that helps us model and understand these relationships mathematically.
The slope of the line with the equation y − 3 = − 2 1 ( x − 2 ) is − 2 1 . This slope indicates that for every 2 units you move to the right (positive x-direction), the line moves down 1 unit (negative y-direction). It represents a negative slope, meaning the line goes downwards as you move from left to right.
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