Slope-Intercept Form: Find The Line Equation

Hey guys! Today, we're diving into how to find the equation of a line when you're given two points it passes through. We'll be using the slope-intercept form, which is a super handy tool in coordinate geometry. So, grab your pencils and let's get started!

Understanding Slope-Intercept Form

The slope-intercept form is a way to represent the equation of a line. It's written as y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis). This form is incredibly useful because it tells you two key things about the line right away: how steep it is (the slope) and where it intersects the y-axis. When you have a line, the slope (m) represents the steepness and direction of the line. It tells you how much the y-value changes for every unit change in the x-value. A positive slope means the line goes upwards from left to right, while a negative slope means the line goes downwards. The y-intercept (b) is the point where the line crosses the y-axis. It's the value of y when x is 0. Knowing the y-intercept gives you a fixed point on the line to start with. The beauty of the slope-intercept form is its simplicity and directness. Once you know the slope and y-intercept, you can easily write the equation of the line and graph it. It's a fundamental concept in algebra and is used extensively in various mathematical and real-world applications. You can determine the slope and y-intercept from the graph or calculate them using the coordinates of two points on the line. Once you have these values, plug them into the equation y = mx + b, and you're all set! The slope-intercept form is a foundational concept in algebra and is essential for understanding linear equations and their graphs. Mastering this form will help you solve a wide range of problems and build a strong foundation for more advanced math topics. The slope-intercept form isn't just a mathematical concept; it has practical applications in various fields. For example, you can use it to model linear relationships in economics, physics, and engineering. Understanding this form allows you to analyze and predict trends, make informed decisions, and solve real-world problems. So, whether you're a student learning the basics of algebra or a professional applying mathematical concepts in your field, the slope-intercept form is a valuable tool to have in your toolkit.

Finding the Slope (m)

The slope (m) is the measure of how much the line rises or falls for every unit of horizontal change. Given two points, (x₁, y₁) and (x₂, y₂), the formula to calculate the slope is: m = (y₂ - y₁) / (x₂ - x₁). This formula calculates the change in y divided by the change in x, which gives you the rate of change of the line. The slope is a crucial characteristic of a line, as it determines its steepness and direction. A positive slope indicates that the line is increasing (going upwards from left to right), while a negative slope indicates that the line is decreasing (going downwards from left to right). A slope of zero means the line is horizontal, and an undefined slope means the line is vertical. Understanding the slope is essential for interpreting the behavior of linear relationships and making predictions based on them. You can determine the slope of a line from its graph by selecting two points on the line and applying the formula. Alternatively, if you have the equation of the line in slope-intercept form (y = mx + b), the slope is simply the coefficient of x. The slope is used in various applications, such as calculating the rate of change of a quantity, determining the steepness of a hill, or modeling linear relationships in data analysis. Being able to calculate and interpret the slope is a fundamental skill in algebra and calculus. The slope isn't just a number; it represents a rate of change. In real-world scenarios, the slope can tell you how quickly something is increasing or decreasing. For example, if you're tracking the sales of a product over time, the slope of the line representing the sales data would tell you how much the sales are increasing or decreasing per unit of time. This information can be invaluable for making business decisions and predicting future trends. Understanding the slope allows you to analyze and interpret data effectively and make informed decisions based on the trends you observe. The concept of slope extends beyond simple lines and is used in calculus to describe the rate of change of curves. The derivative of a function at a point represents the slope of the tangent line to the curve at that point. This concept is essential for understanding optimization problems, where you want to find the maximum or minimum value of a function. So, the slope is a fundamental concept in mathematics with applications in various fields, from basic algebra to advanced calculus.

In our case, we have the points (-6, 1) and (3, 4). Let's plug these into the formula:

m = (4 - 1) / (3 - (-6))

m = 3 / (3 + 6)

m = 3 / 9

m = 1/3

So, the slope of our line is 1/3. This means that for every 3 units we move to the right on the graph, we move 1 unit up.

Finding the Y-Intercept (b)

Now that we have the slope, we need to find the y-intercept (b). We can use the slope-intercept form, y = mx + b, and plug in the slope we just found (m = 1/3) and one of the points, say (-6, 1), to solve for b. The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is 0. Knowing the y-intercept gives you a fixed point on the line to start with. The y-intercept is an important characteristic of a line because it tells you where the line intersects the y-axis. This point is often significant in real-world applications, as it can represent an initial value or a starting point. For example, if you're modeling the growth of a population over time, the y-intercept would represent the initial population size. The y-intercept can be found in several ways. If you have the equation of the line in slope-intercept form (y = mx + b), the y-intercept is simply the constant term b. Alternatively, if you have the graph of the line, you can read the y-intercept directly from the graph by finding the point where the line crosses the y-axis. If you have two points on the line, you can use the slope-intercept form and solve for b, as we'll demonstrate in this article. The y-intercept is used in various applications, such as determining the initial value of a quantity, finding the starting point of a process, or modeling linear relationships in data analysis. Being able to find and interpret the y-intercept is a fundamental skill in algebra and is essential for understanding the behavior of linear relationships. The y-intercept isn't just a point on the graph; it represents an initial condition or a starting value. In many real-world scenarios, the y-intercept has a meaningful interpretation. For example, if you're tracking the cost of a service over time, the y-intercept might represent the initial setup fee. Understanding the significance of the y-intercept can provide valuable insights into the context of the problem. The y-intercept is also closely related to the concept of linear transformations. When you change the y-intercept of a line, you're essentially shifting the line vertically. This transformation can be used to model changes in the initial conditions of a system or to adjust the starting point of a process. So, the y-intercept is a fundamental concept in algebra with applications in various fields, from basic graphing to advanced modeling.

Here's how we do it:

1 = (1/3) * (-6) + b

1 = -2 + b

Now, add 2 to both sides to solve for b:

1 + 2 = b

b = 3

So, the y-intercept is 3. This means the line crosses the y-axis at the point (0, 3).

Putting It All Together

Now that we have both the slope (m = 1/3) and the y-intercept (b = 3), we can write the equation of the line in slope-intercept form:

y = (1/3)x + 3

So, the equation of the line that passes through the points (-6, 1) and (3, 4) is y = (1/3)x + 3.

Conclusion

And there you have it! By using the slope-intercept form, we were able to find the equation of the line that passes through the given points. Remember, the slope-intercept form is a powerful tool that makes it easy to understand and graph linear equations. Keep practicing, and you'll become a pro in no time!

So, the correct answer is:

C. y = (1/3)x + 3