What is the equation of the line that is parallel to the given line and passes through the given point?
$y=-2$
$x=-2$
$y=-4$
$x=-4$
Answer (2)
If the line is parallel to y = − 2 , then the equation is y = − 4 .
If the line is parallel to x = − 2 , then the equation is x = − 4 .
Therefore, the possible equations are y = − 4 and x = − 4 .
The final answer is y = − 4 or x = − 4 .
Explanation
Understanding the Problem We are given two lines: y = − 2 and x = − 2 . We are also given a point which can be interpreted as P ( − 4 , − 4 ) . The goal is to find the equation of a line that is parallel to one of the given lines and passes through the given point.
Case 1: Parallel to y = -2 Let's consider the case where the line we are looking for is parallel to the line y = − 2 . Lines parallel to y = − 2 have the form y = c , where c is a constant. Since the line passes through the point P ( − 4 , − 4 ) , we can substitute the y -coordinate of the point into the equation to find the value of c . Thus, − 4 = c , and the equation of the line is y = − 4 .
Case 2: Parallel to x = -2 Now, let's consider the case where the line we are looking for is parallel to the line x = − 2 . Lines parallel to x = − 2 have the form x = c , where c is a constant. Since the line passes through the point P ( − 4 , − 4 ) , we can substitute the x -coordinate of the point into the equation to find the value of c . Thus, − 4 = c , and the equation of the line is x = − 4 .
Final Answer Therefore, the possible equations for the line are y = − 4 and x = − 4 .
Examples
Imagine you're designing a rectangular garden where one side must align with an existing fence. This problem helps you determine the equations for the other sides of the garden, ensuring they are parallel to the fence and pass through specific points you've marked out. Understanding parallel lines and their equations is crucial in various real-world applications, from architecture and construction to navigation and computer graphics. This algebraic approach ensures precision and alignment in practical tasks.
The equations of the lines that are parallel to the given lines and pass through the point P ( − 4 , − 4 ) are y = − 4 and x = − 4 . These represent a horizontal line and a vertical line respectively. Both lines maintain the parallelism with the original lines provided in the question.
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