Slope-Intercept Form: Find Equation With M = -2 & Point (1,8)

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Slope-Intercept Form: Finding the Equation of a Line

Hey guys! Today, we're going to dive into how to write the equation of a line in slope-intercept form. We'll tackle a specific problem where we're given the slope m=−2{ m = -2 } and a point (1,8){ (1, 8) } that the line passes through. It might sound tricky, but trust me, it's super manageable once you break it down. We're going to go step by step, so you can follow along and really understand the process. Let's get started!

Understanding Slope-Intercept Form

Before we jump into solving the problem, let's quickly recap what slope-intercept form actually is. The slope-intercept form of a linear equation is written as:

y=mx+b{ y = mx + b }

Where:

  • y{ y } is the dependent variable (usually plotted on the vertical axis).
  • x{ x } is the independent variable (usually plotted on the horizontal axis).
  • m{ m } is the slope of the line, representing how steep the line is and its direction (positive or negative).
  • b{ b } is the y-intercept, which is the point where the line crosses the y-axis. It's the value of y{ y } when x=0{ x = 0 }.

Knowing this form is crucial because it gives us a clear picture of the line's characteristics just by looking at the equation. The slope m{ m } tells us about the line's inclination, and the y-intercept b{ b } tells us where the line intersects the y-axis. This makes it incredibly useful for graphing lines and understanding their behavior.

Now, why is understanding slope-intercept form so important? Well, it's like having a secret code to decipher lines. When you see an equation in this form, you instantly know two critical pieces of information about the line: its slope and its y-intercept. This is super handy for various applications, from graphing lines to solving real-world problems involving linear relationships. For instance, think about scenarios like calculating the cost of a service based on an hourly rate (slope) and a fixed fee (y-intercept) or predicting the growth of something over time.

Step 1: Point-Slope Form

The first step in finding the equation of our line is to use the point-slope form. This form is particularly handy when we know a point on the line and the slope. The point-slope form is given by:

y−y1=m(x−x1){ y - y_1 = m(x - x_1) }

Where:

  • (x1,y1){ (x_1, y_1) } is a known point on the line.
  • m{ m } is the slope of the line.

In our case, we have m=−2{ m = -2 } and the point (1,8){ (1, 8) }. So, x1=1{ x_1 = 1 } and y1=8{ y_1 = 8 }. Let's plug these values into the point-slope form:

y−8=−2(x−1){ y - 8 = -2(x - 1) }

This equation, y−8=−2(x−1){ y - 8 = -2(x - 1) }, is the equation of our line in point-slope form. It tells us that the line has a slope of -2 and passes through the point (1, 8). But we're not done yet! We need to convert this into slope-intercept form, which is where the real magic happens.

The point-slope form is a powerful tool because it allows us to write the equation of a line with minimal information. It's like having a map and a compass – you know your starting point (the given point) and your direction (the slope). This form is incredibly versatile and can be used in many different scenarios, making it a must-have in your mathematical toolkit.

Step 2: Simplify to Slope-Intercept Form

Now that we have the equation in point-slope form, y−8=−2(x−1){ y - 8 = -2(x - 1) }, our next step is to simplify it and rewrite it in slope-intercept form y=mx+b{ y = mx + b }. To do this, we need to get y{ y } by itself on one side of the equation. Here’s how we’ll do it:

  1. Distribute the -2: We'll start by distributing the -2 on the right side of the equation: y−8=−2x+2{ y - 8 = -2x + 2 }

  2. Isolate y: Next, we want to get y{ y } alone, so we'll add 8 to both sides of the equation: y−8+8=−2x+2+8{ y - 8 + 8 = -2x + 2 + 8 } y=−2x+10{ y = -2x + 10 }

And there we have it! The equation of the line in slope-intercept form is y=−2x+10{ y = -2x + 10 }. This tells us that the line has a slope of -2 and a y-intercept of 10.

The process of simplifying from point-slope form to slope-intercept form is a fundamental skill in algebra. It's like translating a sentence from one language to another – you're expressing the same information in a different way. This skill is essential for graphing lines, solving systems of equations, and understanding linear relationships in various contexts.

Final Answer

So, to recap, we started with a slope of m=−2{ m = -2 } and a point (1,8){ (1, 8) }. We used the point-slope form to write the equation as y−8=−2(x−1){ y - 8 = -2(x - 1) }, and then we simplified it to slope-intercept form, which gave us y=−2x+10{ y = -2x + 10 }. This is our final answer!

The equation y=−2x+10{ y = -2x + 10 } is a powerful representation of our line. It tells us everything we need to know: the line slopes downwards (because the slope is negative), it's fairly steep (the slope is -2), and it crosses the y-axis at the point (0, 10). Being able to find and interpret these equations is a key skill in algebra and beyond.

Key Takeaways

Let's highlight some key takeaways from this exercise:

  1. Point-Slope Form: This is your best friend when you have a point and a slope. It's like a secret formula that unlocks the equation of a line.
  2. Slope-Intercept Form: This form is super useful for quickly identifying the slope and y-intercept of a line. It's like having a decoder ring for linear equations.
  3. Simplifying Equations: Being able to manipulate equations and rewrite them in different forms is a critical skill in algebra. It's like being a master chef who can transform basic ingredients into a gourmet meal.
  4. Understanding the Connection: Knowing how to move between point-slope form and slope-intercept form gives you a deeper understanding of linear equations. It's like seeing the big picture and all the little details at the same time.

By mastering these concepts, you'll be well-equipped to tackle all sorts of problems involving linear equations. Keep practicing, and you'll become a pro in no time!

Practice Makes Perfect

Now that we've walked through this example, it's your turn to shine! Try working through similar problems on your own. Here are a few suggestions:

  1. Find the equation of the line with a slope of 3 that passes through the point (2, 5).
  2. Find the equation of the line with a slope of -1/2 that passes through the point (-4, 1).
  3. Find the equation of the line that passes through the points (0, -3) and (2, 1).

Remember, the key is to practice and apply the steps we've discussed. The more you work with these concepts, the more comfortable you'll become. So grab a pencil and paper, and let's get those equations flowing!