Which equation represents a line that passes through $(5,1)$ and has a slope of $\frac{1}{2}$?
A. $y-5=\frac{1}{2}(x-1)$
B. $y-\frac{1}{2}=5(x-1)$
C. $y-1=\frac{1}{2}(x-5)$
D. $\left.y-1=5^{\prime} x-\frac{1}{2} \right\rvert\,$
Answer (2)
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 5 , 1 ) and slope 2 1 into the point-slope form: y − 1 = 2 1 ( x − 5 ) .
Compare the resulting equation with the given options.
The correct equation is: y − 1 = 2 1 ( x − 5 )
Explanation
Understanding the Problem We are given a point ( 5 , 1 ) and a slope m = 2 1 . We need to find the equation of the line that passes through this point and has this slope.
Using Point-Slope Form The point-slope form of a linear equation is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Substituting Values Substitute the given point ( 5 , 1 ) and slope m = 2 1 into the point-slope form: y − 1 = 2 1 ( x − 5 )
Comparing with Options Now, we compare this equation with the given options:
Option 1: y − 5 = 2 1 ( x − 1 ) Option 2: y − 2 1 = 5 ( x − 1 ) Option 3: y − 1 = 2 1 ( x − 5 ) Option 4: y − 1 = 5 ( x − 2 1 )
We can see that Option 3 matches the equation we derived using the point-slope form.
Final Answer Therefore, the equation that represents a line that passes through ( 5 , 1 ) and has a slope of 2 1 is: y − 1 = 2 1 ( x − 5 )
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, if you're tracking the distance you travel over time at a constant speed, the relationship can be modeled using a linear equation. If you know your starting point and your speed (slope), you can predict your location at any given time. Similarly, in business, linear equations can help model costs, revenue, and profit, allowing for informed decision-making and forecasting.
The equation representing a line that passes through the point (5, 1) and has a slope of 2 1 is y − 1 = 2 1 ( x − 5 ) , which corresponds to Option C. This equation is derived using the point-slope form of a linear equation. Therefore, the correct answer is Option C.
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