What is the equation of the line parallel to the given line with an $x$-intercept of 4?
$y=\square x+\square$
Answer (2)
The line is parallel to a given line, so it has the same slope m .
The line has an x-intercept of 4, meaning it passes through the point (4, 0).
Using the point-slope form, the equation is y − 0 = m ( x − 4 ) .
Simplifying, the equation of the line is y = m x − 4 m .
Explanation
Understanding the Problem We are given the problem of finding the equation of a line that is parallel to a given line and has an x-intercept of 4. This means the line passes through the point (4, 0). We need to find the equation in the form y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope Since the line is parallel to the given line, it has the same slope. Let's denote the slope of the given line as m . Therefore, the slope of the line we are looking for is also m .
Using Point-Slope Form We know that the line passes through the point (4, 0). We can 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. Substituting the point (4, 0) into the point-slope form, we get y − 0 = m ( x − 4 ) .
Simplifying the Equation Simplifying the equation, we have y = m ( x − 4 ) , which can be written as y = m x − 4 m . This is in the slope-intercept form y = m x + b , where the y-intercept b = − 4 m .
Final Equation Therefore, the equation of the line is y = m x − 4 m .
Examples
Imagine you are designing a ramp for a skateboard park. You need the ramp to be parallel to an existing slope for a smooth transition, and you want the ramp to start at a specific point on the ground (x-intercept). By understanding parallel lines and their equations, you can calculate the exact angle and starting point needed for your ramp to ensure it meets the park's design requirements. This ensures a safe and fun experience for skateboarders.
The equation of the line parallel to the given line with an x-intercept of 4 is y = m x − 4 m , where m is the slope of the given line. This uses the point-slope format based on the known point (4, 0). Ultimately, the slope remains the same for both lines as they are parallel.
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