Solve Linear Functions: A Step-by-Step Guide

Hey guys! Ever stared at a point-slope equation and felt like you were trying to read hieroglyphics? Don't worry, we've all been there. Linear functions might seem intimidating at first, but they're actually super straightforward once you break them down. Today, we're going to tackle a specific problem: finding the linear function that represents the line given by the point-slope equation y + 1 = -3(x - 5). We'll walk through the process step-by-step, so you'll not only get the answer but also understand the why behind it. Let's dive in!

Understanding Point-Slope Form

Before we jump into solving our equation, let's quickly recap what point-slope form actually means. The point-slope form of a linear equation is a way to express the equation of a line using a point on the line and its slope. The general form looks like this: y - y₁ = m(x - x₁), where:

• m is the slope of the line.
• (x₁, y₁) is a specific point on the line.

Think of the slope, m, as the line's steepness and direction. A positive slope means the line goes up as you move from left to right, while a negative slope means it goes down. The point (x₁, y₁) gives us a fixed location that the line passes through. Knowing these two pieces of information – a point and the slope – we can completely define a line.

Now, why is point-slope form so useful? Well, it's incredibly handy when you're given a point and a slope and need to write the equation of the line. It's also a great starting point for converting to other forms, like slope-intercept form, which we'll see in action shortly. In our case, the given equation y + 1 = -3(x - 5) is already in point-slope form. Can you identify the slope and the point just by looking at it? The slope, m, is clearly -3. And the point (x₁, y₁)? Remember the general form has subtraction, so y + 1 is the same as y - (-1), and x - 5 is already in the correct format. That means our point is (5, -1). See, we've already extracted valuable information from the equation!

Transforming to Slope-Intercept Form

The million-dollar question now is: How do we transform the point-slope equation into a linear function in the form f(x) = mx + b? This form, known as slope-intercept form, is another common way to represent linear equations. The beauty of slope-intercept form is that it explicitly shows the slope (m) and the y-intercept (b) – the point where the line crosses the y-axis. Our goal is to manipulate the given equation y + 1 = -3(x - 5) until it looks like f(x) = mx + b. Let’s break down the process step-by-step:

1. Distribute the slope: The first step is to get rid of the parentheses by distributing the -3 across the terms inside: y + 1 = -3 * x + (-3) * (-5). This simplifies to y + 1 = -3x + 15. We're one step closer to isolating y!
2. Isolate y: Our next mission is to get y by itself on the left side of the equation. To do this, we need to get rid of the +1. The opposite of adding 1 is subtracting 1, so we'll subtract 1 from both sides of the equation to maintain balance: y + 1 - 1 = -3x + 15 - 1. This simplifies beautifully to y = -3x + 14. Look at that! We've transformed the equation into slope-intercept form!
3. Express as f(x): Finally, to express this as a linear function, we simply replace y with f(x). So, our final linear function is f(x) = -3x + 14. And there you have it! We've successfully transformed the point-slope equation into slope-intercept form and expressed it as a linear function. This process of distributing and isolating the y variable is fundamental in algebra, and mastering it will make you a linear function whiz!

Identifying the Correct Option

Now that we've done the hard work of transforming the equation, identifying the correct answer from the given options is a piece of cake. We found that the linear function is f(x) = -3x + 14. Let's look at the options:

A. f(x) = -3x - 6 B. f(x) = -3x - 4 C. f(x) = -3x + 16 D. f(x) = -3x + 14

Bingo! Option D, f(x) = -3x + 14, perfectly matches our result. So, the correct answer is D. See how breaking down the problem into smaller, manageable steps made it much easier to solve? We didn't just guess; we understood the process and arrived at the answer confidently. This approach is key to tackling any math problem, no matter how daunting it may seem at first.

Why Other Options Are Incorrect

It's always a good idea to understand not just why the correct answer is right, but also why the other options are wrong. This helps solidify your understanding of the concepts and prevents you from making similar mistakes in the future. Let's analyze why options A, B, and C are incorrect:

• Option A: f(x) = -3x - 6. This option has the correct slope (-3), but the y-intercept is incorrect. This indicates an error in the constant term calculation during the transformation process. Perhaps there was a mistake when adding or subtracting to isolate y.
• Option B: f(x) = -3x - 4. Similar to option A, this also has the correct slope but an incorrect y-intercept. This suggests a different arithmetic error in calculating the constant term. It's possible that the student might have incorrectly combined the constants after distributing the slope.
• Option C: f(x) = -3x + 16. This option also gets the slope right but has a y-intercept that's off. The error here could stem from an incorrect addition or subtraction when isolating y. It's crucial to double-check these steps to ensure accuracy.

By understanding the potential errors that lead to these incorrect options, you can develop a more robust problem-solving approach. Always take the time to review your work and make sure each step is logically sound. Math is like building a house; each step relies on the previous one, so accuracy is paramount.

Key Takeaways and Practice Tips

Alright, guys, we've covered a lot in this explanation! Let's recap the key takeaways and some practice tips to help you master linear functions:

• Point-slope form is your friend: It's a powerful tool for writing the equation of a line when you know a point and the slope. Get comfortable identifying the slope and point directly from the equation.
• Transforming is key: Mastering the transformation from point-slope to slope-intercept form (and vice versa) is crucial. Practice distributing and isolating variables until it becomes second nature.
• Pay attention to signs: A small sign error can throw off your entire answer. Double-check your addition, subtraction, multiplication, and division, especially when dealing with negative numbers.
• Understand the forms: Know the strengths of each form (point-slope and slope-intercept). Slope-intercept form makes it easy to identify the slope and y-intercept, while point-slope form is useful when you have a point and a slope.

Practice Tips:

• Work through similar problems: Find more examples of converting point-slope equations to slope-intercept form. The more you practice, the more confident you'll become.
• Graph the lines: Visualizing the lines can help you understand the relationship between the equation and the graph. Use graphing paper or online tools to plot the lines.
• Check your answers: Always double-check your work, especially the arithmetic. If possible, use a different method to solve the problem and see if you get the same answer.

Linear functions are a fundamental concept in algebra and beyond. By understanding the principles and practicing regularly, you'll build a strong foundation for more advanced math topics. Keep up the great work, and don't hesitate to ask for help when you need it!