Convert Point-Slope Form To Standard Form: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a common algebra problem: converting an equation from point-slope form to standard form. We'll break down the process step-by-step, making it super easy to understand. Don't worry, it's not as scary as it sounds! We'll be using a specific example to illustrate the concept and make sure you understand every single step. So, grab your notebooks, and let's get started! The core concept is transforming an equation from its point-slope representation to the standard form. This is something you'll encounter quite often in your math journey.

Understanding the Problem: From Point-Slope to Standard Form

So, what exactly are we dealing with? We're given the point-slope form of a line's equation and we're asked to find its standard form. Let's clarify what these forms are, just in case you need a refresher. The point-slope form is a way to express a linear equation using a point on the line and its slope. It's generally written as y - y₁ = m(x - x₁), where m is the slope, and (x₁, y₁) is a point on the line. The standard form of a linear equation is usually written as Ax + By = C, where A, B, and C are integers, and A is usually positive. This is the form we're aiming for. Think of it as a way of organizing the equation in a neat, consistent manner. It's all about rearranging the equation to fit this specific structure. Remember, the goal is to manipulate the equation so it matches this Ax + By = C format. Now, let's get into the nitty-gritty of how to convert between the two forms.

The Point-Slope Form: Our Starting Point

The problem provides us with the equation in point-slope form: y + 7 = -2/5(x - 10). Notice that the equation provided, y + 7 = -2/5(x - 10), gives us the slope directly (m = -2/5) and a point on the line (10, -7). This is our starting point. This is the point-slope form. Our job is to manipulate this equation to fit the standard form, Ax + By = C. Before we begin, let's make sure we're all on the same page. We have the slope, and we have a point; what more could we possibly need? Well, we need to get it into that Ax + By = C format. Essentially, we're going to perform algebraic operations to rewrite the equation in this specific arrangement. Don't worry; we will get through this together, nice and easy.

Step-by-Step Conversion: From Point-Slope to Standard

Here's how to convert the point-slope form to the standard form step-by-step. First, we'll deal with the fraction. Then, we'll rearrange terms. Trust me, each step is super important! This methodical approach will ensure we don't miss anything. Let's start with the distribution step.

Step 1: Distribute the Slope

The first thing to do is to distribute the slope (-2/5) to both terms inside the parentheses. This means multiplying -2/5 by both x and -10. So, the equation y + 7 = -2/5(x - 10) becomes: y + 7 = (-2/5)x + 4. It's like giving a gift to both x and -10. That is what you are doing when you distribute. This simplifies the equation and brings us closer to our goal. Make sure you multiply correctly; a small mistake here can mess up the whole process!

Step 2: Eliminate the Fraction

Next, to eliminate the fraction, let's multiply both sides of the equation by 5. Doing this cancels out the denominator and makes the equation easier to work with. Multiplying the whole equation by 5, we get: 5(y + 7) = 5((-2/5)x + 4). Simplifying, this gives us 5y + 35 = -2x + 20.

Step 3: Rearrange the Equation

Now, we need to rearrange the equation to match the standard form Ax + By = C. This means moving all the terms with x and y to one side and the constant term to the other side. Add 2x to both sides, and subtract 35 from both sides. The equation 5y + 35 = -2x + 20 becomes 2x + 5y = -15. You can see how the variables have now been moved to one side. This is the standard form of the equation. It is now in its final and easiest format for us to see.

Step 4: Verify the Solution

At this stage, we've successfully converted the equation to standard form: 2x + 5y = -15. To make sure everything is correct, substitute one of the original points, such as (10, -7), into the equation. If it checks out, you're good to go! Substituting these values, we get: 2*(10) + 5*(-7) = 20 - 35 = -15. The equation holds true, so we have a valid solution. This also means that we have found the answer!

Choosing the Correct Answer

So, now that we've found the standard form of the equation (2x + 5y = -15), we can check the multiple-choice options. In the question, the correct answer is option D: 2x + 5y = -15. Always double-check your answer with the options to make sure you select the correct one. And there you have it! We have successfully converted the point-slope form of the line's equation to its standard form and, moreover, found the correct answer to the given question.

Key Takeaways and Final Thoughts

Converting between the point-slope and standard forms of a linear equation is a fundamental skill in algebra. Remember these key points:

• Understanding the Forms: Know the definitions of point-slope (y - y₁ = m(x - x₁)) and standard form (Ax + By = C).
• Distribute and Simplify: The first step is always to distribute the slope.
• Eliminate Fractions: Multiply to get rid of any fractions.
• Rearrange Terms: Move terms to fit the standard form.
• Verify: Always check your answer by substituting a point into the final equation.

By following these steps, you can confidently convert any point-slope form equation to standard form. It's all about taking it one step at a time. If you practice these steps regularly, you'll master them in no time! Keep practicing, and don't be afraid to ask for help. You've got this! You now have the ability to solve such problems. Congratulations!