Find The Slope-Intercept Equation Of A Line

Understanding how to write the equation of a line in slope-intercept form is a fundamental skill in mathematics. This form, y = mx + b, is incredibly useful because it directly tells you the slope (m) and the y-intercept (b) of the line. This makes it easy to graph the line and understand its behavior. In this article, we’ll walk through the process of finding the slope-intercept form of a line when you’re given a point it passes through and its slope. This is a common scenario in algebra and is essential for solving more complex problems. We'll break down the steps, explain the reasoning behind them, and provide a clear example to solidify your understanding. Whether you're a student learning this for the first time or someone needing a refresher, this guide will help you master the art of writing linear equations.

Understanding Slope-Intercept Form

Let's dive a little deeper into what makes the slope-intercept form (y = mx + b) so special. The slope (m) represents the rate of change of the line. It tells you how much the y-value changes for every one unit increase in the x-value. A positive slope means the line rises from left to right, while a negative slope means it falls. The steeper the line, the larger the absolute value of the slope. The y-intercept (b) is the point where the line crosses the y-axis. At this point, the x-value is always 0. Knowing these two values, m and b, gives you a complete picture of the line's position and direction on a coordinate plane. The beauty of this form is its directness; you can immediately identify key characteristics of the line without any complex calculations. For instance, a line with the equation y = 2x + 3 has a slope of 2 and a y-intercept of 3. This means for every step you take to the right, the line goes up two steps, and it crosses the y-axis at the point (0, 3). This visual and intuitive understanding is why slope-intercept form is so widely used in various mathematical and scientific applications, from analyzing trends in data to modeling physical phenomena. It's the bedrock upon which much of our understanding of linear relationships is built.

The Given Information

In this specific problem, we are provided with two crucial pieces of information that will allow us to determine the unique equation of our line. First, we know a point that the line passes through. This point is given as (-4, -5). This means that when x = -4, y must equal -5 for this line. This specific coordinate pair satisfies the equation of the line. Second, we are given the slope of the line, which is 3/2. This tells us the rate at which the line is changing. For every 2 units we move to the right along the x-axis, the line will move up 3 units along the y-axis. The slope being positive indicates that the line will ascend as we move from left to right. With these two pieces of information – a point and the slope – we have enough to define a single, specific line. If we were only given the slope, there would be infinitely many lines parallel to each other, all with that same slope, but each crossing the y-axis at a different point. Similarly, if we were only given a point, there would be an infinite number of lines that could pass through that single point, each with a different slope. It's the combination of a specific point and a specific slope that locks in the unique position and orientation of the line on the coordinate plane. This is why these two elements are so vital when asked to find the equation of a line.

Steps to Find the Equation

To find the equation of a line in slope-intercept form (y = mx + b), we need to determine the values of m (slope) and b (y-intercept). We are already given the slope, m = 3/2. So, our task is reduced to finding the y-intercept, b. We can use the given point (-4, -5) and the slope to find b. The general idea is to plug the values of x and y from our point into the slope-intercept equation, along with the known slope, and then solve for b. Here’s a step-by-step breakdown:

1. Identify the slope (m): In this problem, the slope is given as m = 3/2.
2. Identify the coordinates of the given point (x, y): The point is (-4, -5), so x = -4 and y = -5.
3. Substitute the known values into the slope-intercept form (y = mx + b): Replace y with -5, m with 3/2, and x with -4. This gives us the equation: -5 = (3/2)(-4) + b.
4. Solve for the y-intercept (b): Now, we need to isolate b. First, calculate the product of the slope and x: (3/2) * (-4) = -12/2 = -6. So the equation becomes: -5 = -6 + b.
5. Isolate b: To get b by itself, add 6 to both sides of the equation: -5 + 6 = -6 + 6 + b. This simplifies to 1 = b.
6. Write the final equation: Now that we have found the slope (m = 3/2) and the y-intercept (b = 1), we can substitute these values back into the slope-intercept form y = mx + b. The equation of the line is y = (3/2)x + 1.

This systematic approach ensures that we use all the given information correctly and logically arrive at the unique equation of the line. Each step builds upon the previous one, making the process manageable and accurate.

Applying the Formula

Let's put the steps into action with our specific problem: finding the equation of a line that passes through the point (-4, -5) and has a slope of 3/2. We'll use the slope-intercept form, y = mx + b, as our foundation. We already know that m = 3/2. Our primary goal now is to find the value of b, the y-intercept. We can achieve this by substituting the coordinates of the given point into the equation. So, we’ll plug in x = -4 and y = -5.

The equation starts as: y = mx + b

Substitute the values: -5 = (3/2)(-4) + b

Now, let's simplify the multiplication of the slope and the x-coordinate: (3/2) * (-4) = -12/2 = -6

Our equation now looks like this: -5 = -6 + b

To solve for b, we need to isolate it. We can do this by adding 6 to both sides of the equation: -5 + 6 = -6 + 6 + b

This simplifies to: 1 = b

So, we have successfully found our y-intercept! It is b = 1. Now that we have both the slope (m = 3/2) and the y-intercept (b = 1), we can write the final equation of the line in slope-intercept form. We simply substitute these values back into the y = mx + b formula.

The final equation is: y = (3/2)x + 1

This equation represents the unique line that satisfies both conditions: it passes through the point (-4, -5) and has a slope of 3/2. You can verify this by plugging x = -4 into the equation and checking if you get y = -5. y = (3/2)(-4) + 1 y = -6 + 1 y = -5

It checks out! This confirms that our derived equation is correct.

Verification of the Equation

Once we have found the equation of the line, it's always a good practice to verify that it correctly represents the given conditions. This step ensures accuracy and builds confidence in our solution. We found the equation to be y = (3/2)x + 1. We need to confirm two things: that this equation has a slope of 3/2 and that it passes through the point (-4, -5).

1. Checking the Slope: The slope-intercept form, y = mx + b, is structured such that the coefficient of x is always the slope (m). In our equation, y = (3/2)x + 1, the coefficient of x is 3/2. This matches the given slope, so the first condition is met.

2. Checking the Point: To verify that the line passes through the point (-4, -5), we substitute the x-coordinate (-4) into our equation and check if the resulting y-coordinate is -5.

Let's substitute x = -4 into y = (3/2)x + 1: y = (3/2)(-4) + 1

First, calculate the product of the slope and x: (3/2) * (-4) = -12/2 = -6

Now, substitute this back into the equation: y = -6 + 1

Perform the addition: y = -5

Since the calculation yields y = -5, which is the y-coordinate of our given point, we can confidently say that the line y = (3/2)x + 1 does indeed pass through the point (-4, -5).

Both conditions are satisfied, confirming that y = (3/2)x + 1 is the correct equation for the line described.

Conclusion

Mastering the process of writing the equation of a line in slope-intercept form is a crucial step in your mathematical journey. By understanding the components of y = mx + b, and by systematically applying the given information – a point and the slope – you can confidently determine the unique equation that defines a specific line. We’ve seen how to use the point-slope relationship to solve for the unknown y-intercept (b) and then assemble the final equation. Remember, the key is to substitute the known values into the slope-intercept formula and algebraically solve for the missing piece. This skill is not just an academic exercise; it's a foundational tool used in graphing, analyzing data, and modeling real-world relationships. Keep practicing, and you'll find that finding the equation of a line becomes second nature!

For further exploration and additional resources on linear equations and coordinate geometry, you can visit the Khan Academy. They offer comprehensive explanations and practice problems to enhance your understanding.