Equation Of A Line Parallel To Y=4x+7 Passing Through (4,5)

Hey there, math enthusiasts! Today, we're going to dive into a classic problem in linear equations: finding the equation of a line that passes through a specific point and is parallel to a given line. Specifically, we want to find the equation of a line that goes through the point (4, 5) and is parallel to the line y = 4x + 7. We'll express our final answer using function notation, which is super handy and widely used in mathematics. So, grab your pencils, and let's get started!

Understanding Parallel Lines and Slopes

Before we jump into the calculations, let's quickly review what it means for lines to be parallel. The most important thing to remember is that parallel lines have the same slope. The slope of a line tells us how steep it is and in what direction it's going. The equation y = 4x + 7 is in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. In our case, the slope of the given line is 4. This means any line parallel to y = 4x + 7 will also have a slope of 4. Understanding this concept is crucial, guys, because it's the foundation for solving this problem. We're essentially looking for a line that has the same steepness and direction as y = 4x + 7, but might cross the y-axis at a different point. Think of it like two lanes on a straight highway – they run alongside each other without ever intersecting because they have the same 'slope' or direction. Recognizing that parallel lines share the same slope simplifies our task considerably, allowing us to focus on finding the specific equation that also satisfies the point (4, 5). We're not just guessing here; we're applying a fundamental geometric principle to nail this problem down. So, let's keep this key concept – parallel lines have the same slope – in the forefront as we move on to the next step.

Using the Point-Slope Form

Now that we know the slope of our desired line (which is 4, the same as the given line), we need to find its equation. Since we also have a point that the line passes through (4, 5), the point-slope form is our best friend here. The point-slope form of a linear equation is given by: y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. This form is incredibly useful because it directly incorporates the slope and a point, making it perfect for our scenario. We already know that m = 4 (the slope of the parallel line) and (x1, y1) = (4, 5) (the given point). Plugging these values into the point-slope form, we get: y - 5 = 4(x - 4). This equation represents a line that has a slope of 4 and passes through the point (4, 5). However, it's not quite in the form we want yet. We need to simplify it and rewrite it in slope-intercept form (y = mx + b) and then in function notation. But the hard part is done! We've successfully used the point-slope form to capture the essence of our line. This form is a powerful tool in your mathematical arsenal, guys, so make sure you're comfortable using it. It's all about plugging in the right values and letting the algebra do its magic. Remember, the point-slope form is not just a formula; it's a way to express a line's equation directly from its slope and a point it passes through, making it an indispensable tool for tackling problems like this one. So, let's move on and simplify this equation to get our final answer.

Converting to Slope-Intercept Form

We've got our equation in point-slope form: y - 5 = 4(x - 4). Now, let's transform it into the familiar slope-intercept form, y = mx + b. This form is super useful because it clearly shows us the slope (m) and the y-intercept (b) of the line. To do this, we need to simplify the equation by distributing the 4 on the right side and then isolating 'y' on the left side. First, distribute the 4: y - 5 = 4x - 16. Next, we want to get 'y' by itself, so we add 5 to both sides of the equation: y = 4x - 16 + 5. Simplifying the right side gives us: y = 4x - 11. Ta-da! We've successfully converted our equation into slope-intercept form. We can now clearly see that the slope (m) is 4, which confirms that our line is indeed parallel to the original line y = 4x + 7. We also know that the y-intercept (b) is -11, meaning the line crosses the y-axis at the point (0, -11). This transformation is a crucial step, guys, because it puts our equation in a standard format that's easy to interpret and compare. The slope-intercept form gives us a clear picture of the line's behavior – its steepness and where it intersects the y-axis. So, by converting to this form, we've not only simplified our equation but also gained valuable insights into the line itself. Now, let's take it one step further and express our equation using function notation, which is the final touch to our problem.

Expressing the Equation in Function Notation

Our equation is currently in slope-intercept form: y = 4x - 11. To express this in function notation, we simply replace 'y' with 'f(x)'. Function notation is a way of writing equations that emphasizes the relationship between the input (x) and the output (y). It's like saying,