Parallel Line Equation: 5x - 4y = 4, Through (-8, 2)

Have you ever wondered how to determine the equation of a line that runs parallel to another, especially when given a specific point it needs to pass through? It's a common challenge in mathematics, and mastering this skill opens doors to understanding more complex geometric concepts. In this guide, we'll break down the process step-by-step, using the example of finding a line parallel to 5x - 4y = 4 and passing through the point (-8, 2). Let's dive in!

Understanding Parallel Lines and Their Slopes

Before we jump into the calculations, let's solidify our understanding of parallel lines. The key concept here is that parallel lines have the same slope. This means they run in the same direction and will never intersect. The slope of a line, often denoted as 'm', represents its steepness and direction. It's calculated as the change in y divided by the change in x (rise over run). Therefore, our initial goal is to find the slope of the given line, 5x - 4y = 4, as any line parallel to it will share this exact slope.

To find the slope, we need to rearrange the equation into slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). This form makes it incredibly easy to identify the slope. So, let’s get started by transforming our given equation.

Step-by-Step: Converting to Slope-Intercept Form

1. Start with the given equation: 5x - 4y = 4
2. Subtract 5x from both sides to isolate the y-term: -4y = -5x + 4
3. Divide both sides by -4 to solve for y: y = (5/4)x - 1

Now we have the equation in slope-intercept form. By examining the equation y = (5/4)x - 1, we can clearly see that the slope, 'm', is 5/4. This is a crucial piece of information because it tells us that any line parallel to 5x - 4y = 4 will also have a slope of 5/4. Remember, the slope represents the steepness and direction of the line, so parallel lines share the same steepness. Next, we'll use this slope and the given point to determine the equation of our parallel line.

Using Point-Slope Form to Find the Equation

Now that we know the slope of our parallel line is 5/4, and we have a point it passes through, (-8, 2), we can utilize the point-slope form of a linear equation. The point-slope form is a powerful tool for constructing the equation of a line when you know its slope and a point on the line. It's expressed as:

**y - y₁ = m(x - x₁) **

Where:

• 'm' is the slope of the line
• **(x₁, y₁) ** is the given point on the line

This form is particularly useful because it directly incorporates the slope and the point, making the process of finding the equation straightforward. Let's plug in our values and see how it works.

Plugging in the Values

We have the slope, m = 5/4, and the point (-8, 2), so x₁ = -8 and y₁ = 2. Substituting these values into the point-slope form, we get:

y - 2 = (5/4)(x - (-8))

Simplifying this equation is our next step. We'll distribute the slope and then isolate 'y' to get the equation in slope-intercept form, which is a more familiar and easily interpretable form.

Simplifying to Slope-Intercept Form

1. Distribute the 5/4 on the right side of the equation: y - 2 = (5/4)x + 10
2. Add 2 to both sides to isolate 'y': y = (5/4)x + 12

Now we have the equation of the parallel line in slope-intercept form: y = (5/4)x + 12. This equation clearly shows the slope (5/4, which is the same as the original line) and the y-intercept (12). However, there's another common form for linear equations called the standard form, which we'll convert to next. Converting to standard form can sometimes make it easier to compare equations and identify certain properties of the line.

Converting to Standard Form

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is typically a positive integer. Converting our equation to standard form involves eliminating the fraction and rearranging the terms. This form is useful for various purposes, such as quickly finding intercepts and comparing different linear equations.

Step-by-Step: Converting to Standard Form

1. Start with the slope-intercept form: y = (5/4)x + 12
2. Subtract (5/4)x from both sides: -(5/4)x + y = 12
3. Multiply both sides by 4 to eliminate the fraction: -5x + 4y = 48
4. Multiply both sides by -1 to make the coefficient of x positive: 5x - 4y = -48

Therefore, the equation of the line parallel to 5x - 4y = 4 and passing through the point (-8, 2) in standard form is 5x - 4y = -48. We now have the equation in both slope-intercept and standard forms, giving us a comprehensive understanding of the line. Knowing how to convert between these forms is a valuable skill in algebra and geometry.

Conclusion: Mastering Parallel Lines

In this guide, we've walked through the process of finding the equation of a line parallel to a given line and passing through a specific point. We started by understanding the fundamental concept that parallel lines share the same slope. We then converted the given equation to slope-intercept form to easily identify the slope. Next, we used the point-slope form to construct the equation of our parallel line, and finally, we converted it to both slope-intercept and standard forms.

This process highlights the interconnectedness of different forms of linear equations and the importance of understanding their properties. By mastering these concepts, you'll be well-equipped to tackle a wide range of mathematical problems involving lines and their relationships. Remember, practice makes perfect, so try working through similar problems to solidify your understanding.

For further exploration of linear equations and their properties, you can visit resources like Khan Academy's Linear Equations for comprehensive lessons and practice exercises.