What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point $(-3,1)$?
A. $y-1=-\frac{3}{2}(x+3)$
B. $y-1=-\frac{2}{3}(x+3)$
C. $y-1=\frac{2}{3}(x+3)$
D. $y-1=\frac{3}{2}(x+3)$
Answer (1)
Identify the slope of the given line as − 2 3 .
Use the fact that parallel lines have the same slope, so the new line also has a slope of − 2 3 .
Plug the point ( − 3 , 1 ) and the slope − 2 3 into the point-slope form y − y 1 = m ( x − x 1 ) .
The equation of the line is y − 1 = − 2 3 ( x + 3 ) .
Explanation
Understanding the Problem We are given a point ( − 3 , 1 ) and need to find the equation of a line that passes through this point and is parallel to a given line. The equation should be in point-slope form, 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. Parallel lines have the same slope.
Finding the Slope The given line is in point-slope form: y − 1 = − 2 3 ( x + 3 ) . We can identify the slope of this line as − 2 3 . Since parallel lines have the same slope, the line we are looking for also has a slope of − 2 3 .
Writing the Equation We are given the point ( − 3 , 1 ) that the line passes through. Using the point-slope form y − y 1 = m ( x − x 1 ) , we can plug in the point ( − 3 , 1 ) and the slope − 2 3 to get the equation of the line: y − 1 = − 2 3 ( x − ( − 3 )) , which simplifies to y − 1 = − 2 3 ( x + 3 ) .
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 the paths never intersect, maintaining safety and efficiency. In architecture, parallel lines are fundamental in creating stable and visually appealing structures. Moreover, in computer graphics and game development, parallel lines are used to create realistic perspectives and spatial relationships between objects.
