Slope-Intercept Form: Find The Equation!

Hey guys! Today, we're diving into a super useful concept in algebra: the slope-intercept form of a line. Specifically, we're going to figure out how to write the equation of a line when we know its slope and a point it passes through. Trust me, once you get the hang of this, you'll be solving these problems like a pro! So, let's break it down step by step. The slope-intercept form is a way to represent the equation of a line, and it's written as y = mx + b. In this equation, 'm' stands for the slope of the line (how steep it is), and 'b' represents the y-intercept (where the line crosses the y-axis). Knowing this form is crucial because it allows us to quickly understand and visualize the line's characteristics.

Understanding Slope-Intercept Form

Let's solidify our understanding of the slope-intercept form, y = mx + b. The slope, denoted by 'm', is the measure of the steepness and direction of a line. It tells us 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 it goes downwards. A slope of zero indicates a horizontal line. The y-intercept, denoted by 'b', is the point where the line intersects the y-axis. It's the y-value when x is zero. Identifying the slope and y-intercept from an equation in slope-intercept form is straightforward, making it a powerful tool for graphing and analyzing linear equations. For instance, if we have the equation y = 2x + 3, we know immediately that the slope is 2 and the y-intercept is 3. This means the line rises 2 units for every 1 unit it moves to the right, and it crosses the y-axis at the point (0, 3). Understanding these components allows us to quickly sketch the line and understand its behavior. Recognizing and utilizing the slope-intercept form is a fundamental skill in algebra, providing a clear and intuitive way to work with linear equations. This form not only simplifies graphing but also aids in solving real-world problems involving linear relationships. So, let’s get comfortable with it!

Applying the Point-Slope Form

Alright, so we're given that the slope (m) is $\frac{1}{3}$ and the line passes through the point (-6, 0). To find the equation of the line, we can use the point-slope form, which is: y - y1 = m(x - x1). Here, (x1, y1) is the given point. Plugging in the values we have, we get: y - 0 = $\frac{1}{3}$(x - (-6)). Simplify this to: y = $\frac{1}{3}$(x + 6). Now, distribute the $\frac{1}{3}$ across the terms inside the parenthesis: y = $\frac{1}{3}$x + $\frac{1}{3}$(6). Simplify further: y = $\frac{1}{3}$x + 2. Boom! We've now converted the equation to slope-intercept form, which is y = mx + b. In our case, m = $\frac{1}{3}$ and b = 2. The point-slope form is particularly useful when you have a point and a slope, as it provides a direct way to construct the equation of the line. By substituting the given values and simplifying, we can easily transform it into the slope-intercept form, which is often more convenient for graphing and analysis. This method ensures accuracy and efficiency in finding the equation of a line, making it an essential tool in algebra. So, remember the point-slope form, and you'll be well-equipped to tackle these types of problems.

Converting to Slope-Intercept Form

To convert the equation we derived from the point-slope form into the slope-intercept form, we need to isolate y on one side of the equation. We started with y = $\frac{1}{3}$(x + 6). Distributing the $\frac{1}{3}$ across the terms inside the parenthesis, we get y = $\frac{1}{3}$x + $\frac{1}{3}$(6). Simplifying the second term, $\frac{1}{3}$(6) becomes 2. So our equation is now y = $\frac{1}{3}$x + 2. This equation is already in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope m is $\frac{1}{3}$ and the y-intercept b is 2. The slope-intercept form is particularly useful because it allows us to quickly identify these key characteristics of the line. From the equation y = $\frac{1}{3}$x + 2, we can immediately see that the line has a slope of $\frac{1}{3}$ and crosses the y-axis at the point (0, 2). This makes it easy to graph the line or compare it to other lines. The process of converting to slope-intercept form involves simplifying the equation to isolate y, which often requires distributing, combining like terms, and performing basic algebraic operations. Mastering this conversion is essential for solving a wide range of problems in algebra and beyond.

Final Answer

So, after going through all the steps, the equation of the line in slope-intercept form is: y = $\frac{1}{3}$x + 2. This matches option D. Therefore, the correct answer is D. y = $\frac{1}{3}$x + 2. Wrapping things up, remember the key steps: identify the slope and point, use the point-slope form, and then convert to slope-intercept form. With practice, you'll nail these problems every time. Keep up the great work, and I'll catch you in the next explanation!