What is the equation of the line parallel to the given line with an [tex]$x$[/tex]-intercept of 4?
[tex]$y=4x+16$[/tex]
Answer (2)
The line parallel to y = 4 x + 16 has the same slope, so its equation is of the form y = 4 x + b .
The x -intercept is 4, meaning the point ( 4 , 0 ) lies on the line.
Substitute ( 4 , 0 ) into y = 4 x + b to find b : 0 = 4 ( 4 ) + b , so b = − 16 .
The equation of the line is y = 4 x − 16 .
Explanation
Understanding the Problem The given line is y = 4 x + 16 . We want to find the equation of a line parallel to this line that has an x -intercept of 4.
Finding the Slope Since the line we are looking for is parallel to y = 4 x + 16 , it will have the same slope. The slope of the given line is 4. Therefore, the equation of the parallel line will be of the form y = 4 x + b , where b is the y -intercept.
Using the x-intercept The x -intercept is the point where the line crosses the x -axis, which means y = 0 . We are given that the x -intercept is 4, so the point ( 4 , 0 ) lies on the line y = 4 x + b .
Solving for the y-intercept Substitute the point ( 4 , 0 ) into the equation y = 4 x + b to solve for b . This gives us:
0 = 4 ( 4 ) + b
0 = 16 + b
b = − 16
Final Equation Therefore, the equation of the line is y = 4 x − 16 .
Examples
Understanding parallel lines and intercepts is crucial in various real-world applications. For instance, consider designing a road parallel to an existing one, where the new road needs to intersect a specific point (the x-intercept) for a connecting route. Similarly, in architecture, parallel lines with defined intercepts are used in creating building plans and ensuring structural stability. This concept also applies in economics when modeling linear relationships between variables, such as cost and production, where understanding intercepts helps in determining break-even points.
The equation of the line parallel to y = 4 x + 16 and having an x -intercept of 4 is y = 4 x − 16 . This line has the same slope of 4 and passes through the point (4, 0).
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