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 (2)
Identify the slope of the given line: m = − 2 3 .
Parallel lines have the same slope, so the new line also has m = − 2 3 .
Use the point-slope form with the point ( − 3 , 1 ) : y − 1 = m ( x + 3 ) .
Substitute the slope to get the equation: 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 must 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 ) . Comparing this to the general point-slope form y − y 1 = m ( x − x 1 ) , we can identify the slope of the given line as m = − 2 3 .
Parallel Line Slope Since the line we want to find is parallel to the given line, it has the same slope. Therefore, the slope of the line we want to find is also m = − 2 3 .
Equation of the Line 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 have x 1 = − 3 and y 1 = 1 . Plugging in the slope m = − 2 3 , we get y − 1 = − 2 3 ( x − ( − 3 )) , which simplifies to y − 1 = − 2 3 ( x + 3 ) .
Final Answer The equation of the line in point-slope form that is parallel to the given line and passes through the point ( − 3 , 1 ) is y − 1 = − 2 3 ( x + 3 ) .
Examples
In architecture, when designing parallel structures like walls or beams, knowing how to find the equation of a line parallel to another is crucial. For instance, if you have a wall represented by a line and you need to build another wall parallel to it at a specific point, this concept helps determine the equation for the new wall's placement. This ensures the structures are aligned and aesthetically pleasing. Understanding parallel lines is also essential in creating blueprints and ensuring structural integrity.
The equation of the line parallel to the given line and passing through the point ( − 3 , 1 ) is y − 1 = − 2 3 ( x + 3 ) . This corresponds to Option A in the multiple-choice question.
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