Find the equation of the line parallel to $y=5 x+9$ that includes the point ( $-2,-3$ ). Give your answer in Point-Slope Form. $y-[?]=\square(x-\square)$
Answer (1)
Determine the slope of the given line: The slope of y = 5 x + 9 is 5.
Use the same slope for the parallel line: The parallel line also has a slope of 5.
Apply the point-slope form: Substitute the point ( − 2 , − 3 ) and slope 5 into y − y 1 = m ( x − x 1 ) .
Write the final equation: The equation is y + 3 = 5 ( x + 2 ) .
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 equation should be in point-slope form. Let's break this down.
Finding the Slope The given line is y = 5 x + 9 . We know that parallel lines have the same slope. The slope of the given line is 5. Therefore, the slope of the line we want to find is also 5.
Using Point-Slope Form We are given the point ( − 2 , − 3 ) that the line passes through. The point-slope form of a line is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Substituting Values Substitute the slope m = 5 and the point ( x 1 , y 1 ) = ( − 2 , − 3 ) into the point-slope form: y − ( − 3 ) = 5 ( x − ( − 2 )) . This simplifies to y + 3 = 5 ( x + 2 ) .
Final Answer So the equation of the line in point-slope form is y + 3 = 5 ( x + 2 ) .
Examples
Imagine you're designing a ramp for a building. You know the slope you need for the ramp to be accessible, and you know it needs to start at a specific point on the ground. Finding the equation of the line (representing the ramp) parallel to a certain slope and passing through a given point is exactly the kind of problem you'd solve in this scenario. This ensures your ramp meets the required slope and starts at the correct location, making it both functional and compliant with accessibility standards.
