Which equation represents a line that passes through $\left(4, \frac{1}{3}\right)$ and has a slope of $\frac{3}{4}$?
A. $y-\frac{3}{4}=\frac{1}{3}(x-4)$
B. $y-\frac{1}{3}=\frac{3}{4}(x-4)$
C. $y-\frac{1}{3}=4\left(x-\frac{3}{4}\right)$
D. $y-4=\frac{3}{4}\left(x-\frac{2}{3}\right)$
Answer (2)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 4 , 3 1 ) and slope 4 3 into the point-slope form: y − 3 1 = 4 3 ( x − 4 ) .
Compare the derived equation with the given options.
The correct equation is y − 3 1 = 4 3 ( x − 4 ) .
Explanation
Understanding the Problem We are given a point ( 4 , 3 1 ) and a slope m = 4 3 . We need to find the equation of the line that passes through this point and has this slope.
Point-Slope Form The point-slope form of a line 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.
Substitution Substitute the given point ( 4 , 3 1 ) and slope 4 3 into the point-slope form: y − 3 1 = 4 3 ( x − 4 )
Comparison Now, we compare this equation with the given options:
Option 1: y − 4 3 = 3 1 ( x − 4 ) Option 2: y − 3 1 = 4 3 ( x − 4 ) Option 3: y − 3 1 = 4 ( x − 4 3 ) Option 4: y − 4 = 4 3 ( x − 3 2 )
We can see that Option 2 matches the equation we derived using the point-slope form.
Examples
Understanding linear equations is crucial in many real-world applications. For example, if you are tracking the distance you travel over time at a constant speed, you can use a linear equation to model this relationship. If you start 2 miles from home and walk at a pace of 3 miles per hour, the equation d = 3 t + 2 represents your distance d from home after t hours. This helps you predict how far you'll be after a certain time or how long it will take to reach a specific distance.
The equation of the line that passes through the point ( 4 , 3 1 ) with a slope of 4 3 is given by y − 3 1 = 4 3 ( x − 4 ) . Therefore, the correct answer is Option B. This matches the standard point-slope form of a linear equation.
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