Which equation represents a line that passes through ($4, \frac{1}{3}$) and has a slope of $\frac{3}{4}$?
A. $y-\frac{3}{4}=\frac{1}{3}(x-4)$
B. $y-\frac{2}{3}=\frac{3}{4}(x-4)$
C. $y-\frac{1}{3}=4\left(x-\frac{3}{4}\right)$
D. $y-4=\frac{3}{4}\left(x-\frac{1}{3}\right)$
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 resulting equation with the given options.
The correct equation is y − 3 1 = 4 3 ( x − 4 ) .
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. 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.
Applying the Point-Slope Form Substitute the given point ( 4 , 3 1 ) and slope 4 3 into the point-slope form: y − 3 1 = 4 3 ( x − 4 ) Now, we compare this equation with the given options to find the correct one.
Comparing with the Options The equation we derived is: y − 3 1 = 4 3 ( x − 4 ) Comparing this with the given options:
y − 4 3 = 3 1 ( x − 4 ) - Incorrect
y − 3 2 = 4 3 ( x − 4 ) - Incorrect
y − 3 1 = 4 ( x − 4 3 ) - Incorrect
y − 4 = 4 3 ( x − 3 1 ) - Incorrect
Only the second option, y − 3 2 = 4 3 ( x − 4 ) , is close, but it has 3 2 instead of 3 1 on the left side. However, the correct equation is: y − 3 1 = 4 3 ( x − 4 ) So, the correct option is: y − 3 1 = 4 3 ( x − 4 )
Final Answer However, none of the options exactly match the equation we derived. Let's check the options again:
y − 4 3 = 3 1 ( x − 4 ) - Incorrect
y − 3 2 = 4 3 ( x − 4 ) - Incorrect
y − 3 1 = 4 ( x − 4 3 ) - Incorrect
y − 4 = 4 3 ( x − 3 1 ) - Incorrect
It seems there was a typo in the options provided. The correct equation should be: y − 3 1 = 4 3 ( x − 4 ) However, if we look at the options again, we can see that option 2 is the closest one. Let's rewrite the equation we derived: y − 3 1 = 4 3 x − 3 y = 4 3 x − 3 + 3 1 y = 4 3 x − 3 9 + 3 1 y = 4 3 x − 3 8 Now let's check if the point ( 4 , 3 1 ) satisfies this equation: 3 1 = 4 3 ( 4 ) − 3 8 3 1 = 3 − 3 8 3 1 = 3 9 − 3 8 3 1 = 3 1 The point satisfies the equation. Now let's look at the options again. The closest one is: y − 3 2 = 4 3 ( x − 4 ) which can be written as y = 4 3 x − 3 + 3 2 = 4 3 x − 3 7 . This is not the correct equation. However, the first option is y − 3 1 = 4 3 ( x − 4 ) .
Final Answer The correct equation is: y − 3 1 = 4 3 ( x − 4 ) So the answer is y − 3 1 = 4 3 ( x − 4 )
Examples
Understanding the equation of a line is crucial in many real-world applications. For instance, in physics, you can use it to describe the motion of an object moving at a constant speed. In economics, you can model the relationship between supply and demand. In computer graphics, lines are fundamental building blocks for creating images and animations. Knowing how to find the equation of a line given a point and a slope allows you to make predictions and solve problems in these diverse fields.
The equation representing the line through the point ( 4 , 3 1 ) with a slope of 4 3 is y − 3 1 = 4 3 ( x − 4 ) . This equation matches the point-slope form of a line. However, none of the given options matched this derived equation directly.
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