Slope-Intercept Form Equation Find Line Equation With Slope 1/4 And Point (0,5)

Hey everyone! Today, we're diving into the wonderful world of linear equations, specifically focusing on the slope-intercept form, which is a super handy way to represent the equation of a line. We'll break down what slope-intercept form is all about and then walk through an example to solidify your understanding. So, let's get started and make math a little less intimidating!

What is Slope-Intercept Form?

The slope-intercept form is a way to write the equation of a line: y = mx + b. In this equation, each letter has a special meaning:

• y: Represents the y-coordinate of any point on the line.
• x: Represents the x-coordinate of any point on the line.
• m: Represents the slope of the line. The slope tells us how steep the line is and in which direction it's going (uphill or downhill).
• b: Represents the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is 0.

Think of the slope (m) as the rate of change of the line. If m is positive, the line goes upwards as you move from left to right. If m is negative, the line goes downwards. The larger the absolute value of m, the steeper the line. The y-intercept (b) is our starting point on the y-axis. It's where the line begins its journey across the coordinate plane. Understanding these components is key to mastering linear equations and their graphical representations. The beauty of slope-intercept form lies in its simplicity and the clear information it provides about the line's characteristics. By simply looking at the equation, we can immediately identify the slope and the y-intercept, which allows us to quickly sketch the line or analyze its behavior. This form is particularly useful in various applications, such as modeling real-world scenarios involving constant rates of change, like the speed of a car or the cost of a service per unit. Moreover, slope-intercept form serves as a foundation for understanding more complex linear concepts and provides a visual and intuitive way to grasp the relationship between variables. So, let's embrace the power of y = mx + b and unlock the secrets it holds for us in the world of linear equations.

Finding the Equation of a Line in Slope-Intercept Form

Now, let's get to the fun part: finding the equation of a line when we're given some information. There are a few different scenarios, but we'll focus on one specific example here: when we know the slope (m) and a point on the line (x, y). This is a common scenario, and the process is quite straightforward.

Here's the general approach:

1. Write down the slope-intercept form: y = mx + b
2. Substitute the given slope (m) into the equation.
3. Substitute the x and y coordinates of the given point into the equation.
4. Solve for b (the y-intercept).
5. Write the final equation by substituting the values of m and b back into the slope-intercept form.

Let's illustrate this process with an example. Suppose we're given a slope of $rac{1}{4}$ and a point on the line (0, 5). This means m = $rac{1}{4}$, x = 0, and y = 5. Now, let's follow the steps:

1. Write down the slope-intercept form: y = mx + b
2. Substitute the slope: y = $rac{1}{4}$x + b
3. Substitute the point (0, 5): 5 = $rac{1}{4}$(0) + b
4. **Solve for b: **
   • 5 = 0 + b
   • b = 5
5. Write the final equation: y = $rac{1}{4}$x + 5

And there you have it! The equation of the line in slope-intercept form is y = $rac{1}{4}$x + 5. This equation tells us that the line has a slope of $rac{1}{4}$ and crosses the y-axis at the point (0, 5). This method is incredibly powerful because it allows us to find the equation of a line even when we don't have the full picture. Knowing just the slope and one point is enough to define the entire line, thanks to the magic of slope-intercept form. By following these steps carefully, you can confidently tackle similar problems and build a strong foundation in linear equations. Remember, practice makes perfect, so don't hesitate to try out more examples and solidify your understanding. The more you work with slope-intercept form, the more intuitive it will become, and you'll be solving linear equation problems like a pro in no time! Understanding the underlying concepts and the logical flow of the steps is crucial for success. So, keep practicing and keep exploring the fascinating world of mathematics!

Example: Finding the Equation with Slope $rac{1}{4}$ and Point (0, 5)

Let's work through the example given in the problem statement. We have a slope (m) of $rac{1}{4}$ and a point (0, 5) on the line. Our goal is to find the equation of the line in slope-intercept form (y = mx + b).

We already know that the slope-intercept form is y = mx + b. We are given that the slope, m, is $rac{1}{4}$. We are also given a point on the line, (0, 5). This means that when x = 0, y = 5. This is super convenient because the point (0, 5) is actually the y-intercept! When x = 0, the y value is the y-intercept (b).

Since we have the point (0, 5), which is the y-intercept, we know that b = 5. We already know m = $rac{1}{4}$. So, we can simply plug these values into the slope-intercept form: y = mx + b.

Substituting the values, we get:

• y = $rac{1}{4}$x + 5

This is the equation of the line in slope-intercept form! This example perfectly demonstrates how slope-intercept form can be derived directly when a slope and y-intercept are provided. Since (0,5) is the point given, we understand it is the y-intercept as x = 0 and y = 5. This tells us that b = 5 and we simply input this into our formula, along with the provided slope. It's a straightforward process once you grasp the fundamental components of the slope-intercept equation, and with repeated practice, it can be easily executed in other contexts. Always ensure that you are attentive to the particulars of the problem so as to best choose the solution pathway, this way, problems become simpler and you can have more confidence in your mathematical abilities. Also, remember that finding the equation of a line is just the beginning, because once you have an equation, there are numerous things you can accomplish, such as finding other points on the line, graphing it, and comprehending its relations with other lines. So keep pushing and revel in the vast opportunities that mathematics presents!

Key Takeaways

• The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
• To find the equation of a line in slope-intercept form, you need to know the slope and the y-intercept.
• If you're given a point other than the y-intercept, you can substitute the slope and the point's coordinates into the slope-intercept form and solve for b.
• Understanding slope-intercept form makes it easier to graph lines and analyze their properties.

Mastering the slope-intercept form is a crucial step in your journey through algebra and beyond. It provides a powerful tool for understanding and representing linear relationships, which are fundamental in many areas of mathematics, science, and engineering. By grasping the concepts of slope and y-intercept and how they relate to the equation of a line, you'll be well-equipped to tackle a wide range of problems and applications. Remember, practice is key, so don't hesitate to work through numerous examples and explore different scenarios. The more you engage with slope-intercept form, the more comfortable and confident you'll become in using it. So, keep up the great work, and happy problem-solving!

I hope this explanation has been helpful! Feel free to ask if you have any more questions.