Slope-Intercept Form How To Find The Equation Of A Line
Hey guys! Let's dive into a classic math problem: finding the slope-intercept form of a line. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems down the road. In this article, we'll walk through a step-by-step solution to a specific problem, and along the way, we'll break down the underlying principles so you can confidently solve similar problems on your own. We'll focus on finding the slope-intercept form of a line that passes through two given points. This involves calculating the slope and then using one of the points to determine the y-intercept. So, let's get started and make math a little less mysterious!
Understanding the Slope-Intercept Form
Before we jump into the problem, let's quickly review the slope-intercept form of a linear equation. The slope-intercept form is a way of writing the equation of a line that makes it easy to identify the line's slope and y-intercept. The general form is:
y = mx + b
Where:
- y represents the dependent variable (typically plotted on the vertical axis).
- x represents the independent variable (typically plotted on the horizontal axis).
- m represents the slope of the line, which indicates its steepness and direction.
- b represents the y-intercept, which is the point where the line crosses the y-axis.
Understanding what each of these variables represents is crucial for interpreting and working with linear equations. The slope, often referred to as "rise over run," 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. The y-intercept, on the other hand, is a specific point on the line – the point where x equals zero. Knowing these two key pieces of information, the slope and the y-intercept, allows us to completely define and graph a straight line.
Now that we've refreshed our understanding of the slope-intercept form, we can move on to the problem at hand. Our goal is to find the equation of a line in this form, given two points that the line passes through. This involves a couple of steps: first, we need to calculate the slope using the coordinates of the two points, and second, we need to use the slope and one of the points to find the y-intercept. Once we have both the slope (m) and the y-intercept (b), we can simply plug them into the y = mx + b equation to get our final answer. So, let's roll up our sleeves and get to work!
Problem Statement
The problem we're tackling today is: A line passes through the points (2, -7) and (-3, 3). Find the slope-intercept form of the equation of the line. Then, we need to identify the value of the slope, m, and the value of the y-intercept, b. This is a common type of problem in algebra, and it's a great way to practice working with linear equations. To solve this, we'll need to use a couple of key formulas and concepts. First, we'll use the slope formula to calculate the slope of the line given two points. Then, we'll use the slope and one of the points in the point-slope form of a line to find the equation of the line. Finally, we'll convert the equation to slope-intercept form, which will allow us to easily identify the values of m and b. It might sound like a few steps, but don't worry, we'll break it down into manageable chunks and explain each step clearly.
So, let's reiterate the problem to make sure we're all on the same page: We have two points, (2, -7) and (-3, 3), and we want to find the equation of the line that passes through both of them. We specifically want the equation in slope-intercept form (y = mx + b), so we need to find the values of m (the slope) and b (the y-intercept). This is a classic example of how algebra can be used to describe geometric relationships, and it's a fundamental skill for anyone studying mathematics or related fields. By the end of this article, you'll have a solid understanding of how to solve this type of problem, and you'll be able to apply the same techniques to other similar problems. So, let's get started and unravel the mystery of finding the slope-intercept form!
Step 1: Calculate the Slope (m)
The first crucial step in finding the slope-intercept form of the equation is calculating the slope (m) of the line. The slope represents the steepness and direction of the line. To calculate the slope, we use the slope formula, which is derived from the concept of "rise over run." The slope formula is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Where:
- (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line.
In our problem, we are given the points (2, -7) and (-3, 3). Let's assign these values to our variables:
- x₁ = 2
- y₁ = -7
- x₂ = -3
- y₂ = 3
Now, we can plug these values into the slope formula:
m = (3 - (-7)) / (-3 - 2)
Simplifying the equation, we get:
m = (3 + 7) / (-5)
m = 10 / -5
m = -2
Therefore, the slope of the line passing through the points (2, -7) and (-3, 3) is -2. A negative slope indicates that the line is decreasing as we move from left to right. This means that for every one unit we move to the right along the x-axis, the y-value decreases by two units. The slope is a fundamental property of a line, and it's essential for understanding its behavior and position on the coordinate plane. Now that we have calculated the slope, we're one step closer to finding the complete equation of the line in slope-intercept form. The next step will involve using this slope and one of the given points to determine the y-intercept, which is the final piece of the puzzle.
Step 2: Find the y-intercept (b)
Now that we've successfully calculated the slope (m) of the line, which is -2, the next step is to find the y-intercept (b). The y-intercept is the point where the line crosses the y-axis, and it's a crucial component of the slope-intercept form of the equation. To find the y-intercept, we can use the slope we just calculated and one of the given points in the point-slope form of a linear equation. The point-slope form is given by:
y - y₁ = m(x - x₁)
Where:
- m is the slope of the line.
- (x₁, y₁) is any point on the line.
We can use either of the points given in the problem, (2, -7) or (-3, 3). Let's choose the point (2, -7) for this calculation. So, we have:
- x₁ = 2
- y₁ = -7
- m = -2
Plugging these values into the point-slope form, we get:
y - (-7) = -2(x - 2)
Simplifying the equation, we get:
y + 7 = -2x + 4
Now, to find the y-intercept, we need to isolate y to get the equation in slope-intercept form (y = mx + b). Subtracting 7 from both sides of the equation, we get:
y = -2x + 4 - 7
y = -2x - 3
Now the equation is in slope-intercept form, and we can easily identify the y-intercept, b. In this case, b is -3. This means that the line crosses the y-axis at the point (0, -3). Finding the y-intercept is a critical step in defining the line, as it tells us where the line is positioned vertically on the coordinate plane. With both the slope and the y-intercept determined, we now have all the information we need to write the complete equation of the line in slope-intercept form. In the next step, we'll put it all together and present the final answer.
Step 3: Write the Slope-Intercept Form
We've done the groundwork! We've calculated the slope (m) and found the y-intercept (b). Now, the final step is to put these values together to write the equation of the line in slope-intercept form. Remember, the slope-intercept form is:
y = mx + b
We found that the slope, m, is -2, and the y-intercept, b, is -3. Now, we simply substitute these values into the equation:
y = (-2)x + (-3)
Simplifying, we get:
y = -2x - 3
This is the slope-intercept form of the equation of the line that passes through the points (2, -7) and (-3, 3). This equation tells us everything we need to know about the line: its steepness (slope of -2) and where it crosses the y-axis (y-intercept of -3). We can even use this equation to graph the line or to find other points that lie on the line. The slope-intercept form is a powerful tool for understanding and working with linear equations, and by following these steps, you can confidently find the equation of any line given two points.
To summarize, we first calculated the slope using the slope formula, then we used the point-slope form to find the equation of the line, and finally, we converted it to slope-intercept form. This process might seem like a lot of steps at first, but with practice, it becomes a straightforward and valuable skill. Now, let's explicitly state the values of m and b as requested in the problem statement.
Final Answer
Okay, let's wrap things up and provide the final answer in the format requested by the problem. We've successfully found the slope-intercept form of the equation of the line, which is:
y = -2x - 3
Now, let's identify the values of the slope, m, and the y-intercept, b:
- m = -2
- b = -3
So, to fill in the blanks:
m = -2 b = -3
And there you have it! We've successfully found the slope-intercept form of the equation of the line and identified the slope and y-intercept. This problem demonstrates a fundamental concept in algebra, and the skills you've learned here will be valuable for tackling more complex problems in the future. Remember, the key is to break down the problem into smaller, manageable steps, and to understand the underlying principles behind each step. With practice and a solid understanding of the concepts, you can confidently solve a wide range of linear equation problems. We hope this step-by-step guide has been helpful, and that you feel more confident in your ability to work with slope-intercept form. Keep practicing, and you'll become a pro in no time!
