What is the slope of the line whose equation is $y-4=\frac{5}{2}(x-2)$?
Answer (2)
Recognize the equation is in point-slope form: y − y 1 = m ( x − x 1 ) .
Identify the slope m as the coefficient of ( x − x 1 ) in the given equation.
The slope of the line y − 4 = 2 5 ( x − 2 ) is 2 5 .
Therefore, the slope is 2 5 .
Explanation
Identifying the Form of the Equation The equation of the line is given as y − 4 = 2 5 ( x − 2 ) . This equation is in point-slope form, which is y − y 1 = m ( x − x 1 ) , where m represents the slope of the line and ( x 1 , y 1 ) is a point on the line.
Determining the Slope Comparing the given equation y − 4 = 2 5 ( 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 ) . In this case, the coefficient is 2 5 .
Conclusion Therefore, the slope of the line is 2 5 .
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 slope value) means a more inclined roof, which can affect water runoff and snow accumulation. 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 larger slope means a greater change in quantity for a given change in price. In construction, architecture, and economics, understanding the concept of slope helps in making informed decisions and predictions.
The slope of the line given by the equation y − 4 = 2 5 ( x − 2 ) is 2 5 . This is determined by identifying the coefficient of ( x − x 1 ) in the point-slope form of the equation. Therefore, the slope is 2 5 .
;
