Find The Equation Of A Parallel Line In Slope-Intercept Form
Let's dive into how to find the equation of a line that's parallel to another line, especially when we need to express it in the slope-intercept form. This is a common task in algebra and geometry, and understanding the steps makes it quite straightforward. We'll start with a given line and a point, and then we'll work our way to the equation of the parallel line.
Understanding the Basics: Slope-Intercept Form and Parallel Lines
Before we jump into the problem, let's quickly recap a couple of key concepts.
- Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis).
- Parallel Lines: Parallel lines are lines in the same plane that never intersect. A crucial property of parallel lines is that they have the same slope. This fact is essential for solving our problem. When we know the slope of the given line, we immediately know the slope of any line parallel to it.
Knowing these basics will help us break down the problem and solve it step by step. We'll use the given information to find the slope of the parallel line and then use the point it passes through to determine its exact equation.
Step-by-Step Solution
Here’s how to find the equation of line s, which is parallel to line r and passes through the point (-1, -1).
1. Identify the Slope of Line r
The equation of line r is given as y + 6 = -3/2(x + 6). To find the slope, we need to rewrite this equation in slope-intercept form (y = mx + b). Let's do that:
y + 6 = -3/2(x + 6)
y + 6 = -3/2x - 9
y = -3/2x - 9 - 6
y = -3/2x - 15
From this, we can see that the slope of line r is -3/2. This is the m value in the slope-intercept form. The slope tells us how steep the line is and whether it's increasing or decreasing. In this case, for every 2 units we move to the right on the graph, we move 3 units down, indicating a decreasing line.
2. Determine the Slope of Line s
Since line s is parallel to line r, it has the same slope. Therefore, the slope of line s is also -3/2. This is a fundamental property of parallel lines: they have identical slopes, ensuring they never intersect. Now that we know the slope of line s, we're halfway to finding its equation.
3. Use the Point-Slope Form
Line s passes through the point (-1, -1). We can use the point-slope form of a linear equation to find the equation of line s. The point-slope form is:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line and m is the slope. In our case, (x₁, y₁) = (-1, -1) and m = -3/2. Plugging these values into the point-slope form, we get:
y - (-1) = -3/2(x - (-1))
y + 1 = -3/2(x + 1)
The point-slope form is useful because it allows us to create the equation of a line knowing just one point and the slope. From here, we'll convert it into the slope-intercept form to get our final answer.
4. Convert to Slope-Intercept Form
Now, let's convert the equation from point-slope form to slope-intercept form (y = mx + b):
y + 1 = -3/2(x + 1)
y + 1 = -3/2x - 3/2
y = -3/2x - 3/2 - 1
y = -3/2x - 3/2 - 2/2
y = -3/2x - 5/2
So, the equation of line s in slope-intercept form is y = -3/2x - 5/2. This equation tells us that the line has a slope of -3/2 and crosses the y-axis at the point (0, -5/2).
5. Final Answer
The equation of line s in slope-intercept form is:
y = -3/2x - 5/2
This is our final answer. We started with the equation of a line r, used the fact that parallel lines have the same slope, and then applied the point-slope form to find the equation of line s. Finally, we converted it into slope-intercept form to clearly show the slope and y-intercept.
Additional Practice and Resources
To solidify your understanding, try working through similar problems. For example, find the equation of a line parallel to y = 2x + 3 that passes through the point (2, 1). The process is the same: identify the slope, use the point-slope form, and convert to slope-intercept form.
Understanding how to find equations of parallel lines is a fundamental skill in algebra and geometry. It reinforces the concepts of slope, intercepts, and linear equations. With practice, you'll become more comfortable and confident in solving these types of problems. Remember, the key is to take it step by step and understand the underlying principles.
Conclusion
Finding the equation of a line parallel to another line involves understanding the properties of parallel lines and using algebraic techniques to manipulate equations. By following a step-by-step approach, we can easily convert the equation into slope-intercept form, which provides valuable information about the line's slope and y-intercept. Remember to practice regularly to reinforce your understanding and improve your problem-solving skills. With a solid grasp of these concepts, you'll be well-equipped to tackle more advanced problems in algebra and geometry.
For more information on linear equations and parallel lines, check out resources like Khan Academy's Linear Equations.
