Identify an equation in point-slope form for the line parallel to [tex]y=-\frac{2}{3} x+8[/tex] that passes through (4,-5).
A. [tex]y+5=-\frac{2}{3}(x-4)[/tex]
B. [tex]y+5=\frac{3}{2}(x-4)[/tex]
C. [tex]y-5=-\frac{2}{3}(x+4)[/tex]
D. [tex]y-4=\frac{2}{3}(x+5)[/tex]
Answer (1)
Determine the slope of the given line: m = − 3 2 .
Parallel lines have the same slope: m = − 3 2 .
Use the point-slope form: y − y 1 = m ( x − x 1 ) .
Substitute the point ( 4 , − 5 ) and slope: y + 5 = − 3 2 ( x − 4 ) . The answer is y + 5 = − 3 2 ( x − 4 ) .
Explanation
Understanding the Problem The problem asks us to find the equation of a line in point-slope form that is parallel to a given line and passes through a specific point. The given line is y = − 3 2 x + 8 , and the point is ( 4 , − 5 ) .
Finding the Slope First, we need to determine the slope of the given line. The equation y = − 3 2 x + 8 is in slope-intercept form, y = m x + b , where m represents the slope. Therefore, the slope of the given line is − 3 2 .
Parallel Lines Have Equal Slopes Since the line we are looking for is parallel to the given line, it will have the same slope. Thus, the slope of the parallel line is also − 3 2 .
Using Point-Slope Form Now, we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. We are given the point ( 4 , − 5 ) , so x 1 = 4 and y 1 = − 5 . We also know that the slope m = − 3 2 .
Substituting Values Substitute the values into the point-slope form: y − ( − 5 ) = − 3 2 ( x − 4 ) . This simplifies to y + 5 = − 3 2 ( x − 4 ) .
Identifying the Correct Option Comparing our equation y + 5 = − 3 2 ( x − 4 ) with the given options, we see that it matches option A.
Final Answer Therefore, the equation of the line in point-slope form that is parallel to y = − 3 2 x + 8 and passes through ( 4 , − 5 ) is y + 5 = − 3 2 ( x − 4 ) .
Examples
Understanding parallel lines and their equations is crucial in various real-world applications. For instance, consider designing roads or railway tracks. Parallel lines ensure that lanes or tracks run alongside each other without intersecting, maintaining a constant distance for safety and efficiency. In architecture, parallel lines are fundamental in creating symmetrical and balanced designs, contributing to the aesthetic appeal and structural integrity of buildings. Moreover, in computer graphics and game development, parallel lines are used to create perspective and depth, enhancing the visual realism of virtual environments.
