What is the equation of the line that is parallel to the line $y=-\frac{1}{3} x+4$ and passes through the point $(6,5)$?
A. $y=-\frac{1}{3} x+3$
B. $y=-\frac{1}{3} x+7$
C. $y=3 x-13$
D. $y=3 x+5$
Answer (2)
The line parallel to y = − 3 1 x + 4 has the same slope, − 3 1 .
Substitute the point ( 6 , 5 ) into the equation y = − 3 1 x + b to find the y-intercept b .
Solve for b : 5 = − 3 1 ( 6 ) + b ⇒ b = 7 .
The equation of the line is y = − 3 1 x + 7 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is parallel to a given line and passes through a specific point. The given line is y = − 3 1 x + 4 , and the point is ( 6 , 5 ) .
Finding the Slope Since the line we are looking for is parallel to y = − 3 1 x + 4 , it will have the same slope. The slope of the given line is − 3 1 . Therefore, the equation of the new line will be of the form y = − 3 1 x + b , where b is the y-intercept we need to find.
Using the Given Point We know that the line passes through the point ( 6 , 5 ) . This means that when x = 6 , y = 5 . We can substitute these values into the equation y = − 3 1 x + b to solve for b .
Solving for the y-intercept Substituting x = 6 and y = 5 into the equation, we get: 5 = − 3 1 ( 6 ) + b 5 = − 2 + b Adding 2 to both sides, we find: b = 7
Writing the Equation of the Line Now that we have the slope − 3 1 and the y-intercept 7 , we can write the equation of the line as: y = − 3 1 x + 7
Final Answer The equation of the line that is parallel to y = − 3 1 x + 4 and passes through the point ( 6 , 5 ) is y = − 3 1 x + 7 .
Examples
Understanding parallel lines is crucial in various real-world applications. For instance, consider city planning where streets are often designed to be parallel to each other. If a new street needs to be constructed parallel to an existing one and pass through a specific location, the principles of parallel lines and their equations are applied to ensure accurate alignment and efficient urban development. Similarly, in architecture, parallel lines are fundamental in designing structures with symmetrical and balanced aesthetics. Knowing how to determine the equation of a line parallel to another helps in creating precise and visually appealing designs.
The equation of the line parallel to y = − 3 1 x + 4 that passes through the point ( 6 , 5 ) is y = − 3 1 x + 7 . The correct answer is option B. This was found by using the parallel slope and substituting the point to find the y-intercept.
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