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

Hey math enthusiasts! Today, we're diving into a common algebra problem: converting an equation from point-slope form to standard form. The specific problem we'll tackle is transforming the equation of a line that passes through the points $(-4, -3)$ and $(12, 1)$, given initially in point-slope form as $y - 1 = \frac{1}{4}(x - 12)$, into its standard form. This process is super important because understanding different forms of linear equations gives you flexibility when solving problems and interpreting the line's properties. Let's break it down step by step so you can ace similar questions with confidence. So, let's get started, guys!

Understanding the Basics: Point-Slope and Standard Form

Before we jump into the conversion, it's crucial to understand the two forms of linear equations we're working with.

Point-Slope Form

As the name suggests, point-slope form uses a point on the line and the slope to define the equation. The general format is: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line, and $m$ is the slope. In our problem, the given equation $y - 1 = \frac{1}{4}(x - 12)$ is already in point-slope form. We can identify that the line has a slope of $\frac{1}{4}$, and it passes through the point $(12, 1)$. Cool, right?

Standard Form

Standard form, on the other hand, is written as $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is typically positive. This form is particularly useful for quickly identifying the x and y intercepts and for solving systems of linear equations. Our goal is to manipulate the point-slope equation to fit this format. This might seem like a lot, but trust me; it's easier than it sounds.

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

Now, let's convert the given equation $y - 1 = \frac{1}{4}(x - 12)$ into standard form. Follow these steps, and you'll be a pro in no time.

Step 1: Distribute the Slope

The first thing we need to do is distribute the slope, $\frac{1}{4}$, across the terms inside the parentheses. This means multiplying $\frac{1}{4}$ by both $x$ and $-12$.

So, $y - 1 = \frac{1}{4}x - \frac{1}{4} \cdot 12$ simplifies to $y - 1 = \frac{1}{4}x - 3$. See? Easy peasy!

Step 2: Eliminate Fractions (If Necessary)

In our case, we have a fraction, which isn't allowed in the standard form (where the coefficients should ideally be integers). To get rid of the fraction, we'll multiply the entire equation by the denominator of the fraction, which is 4. This clears the fraction, making our equation easier to manage. Remember, what we do to one side, we must do to the other to keep the equation balanced. Multiplying the entire equation by 4:

$4(y - 1) = 4(\frac{1}{4}x - 3)$

This simplifies to: $4y - 4 = x - 12$. Now we're getting somewhere!

Step 3: Rearrange the Equation

The final step is to rearrange the equation to match the standard form $Ax + By = C$. We need to move the $x$ term to the left side of the equation and the constant terms to the right side. So, let's subtract $x$ from both sides and add 4 to both sides to get our terms in order.

Starting with $4y - 4 = x - 12$, subtract $x$ from both sides: $-x + 4y - 4 = -12$. Then, add 4 to both sides: $-x + 4y = -8$.

Step 4: Adjust the Coefficient of x (Make 'A' Positive)

In standard form, the coefficient $A$ (the coefficient of $x$) should be positive. Currently, our $x$ coefficient is $-1$. To fix this, we'll multiply the entire equation by $-1$. Doing this changes the sign of every term in the equation.

So, $-1(-x + 4y) = -1(-8)$ becomes $x - 4y = 8$. And there you have it, folks! We've successfully converted our equation into standard form.

Conclusion: The Standard Form Equation

After following these steps, the standard form of the equation $y - 1 = \frac{1}{4}(x - 12)$ is $x - 4y = 8$. Comparing this with the answer choices provided, we can see that option B. $x - 4y = 8$ is the correct answer. Congratulations! You've successfully converted from point-slope to standard form. You guys rock!

Additional Tips and Tricks

Here are some extra tips to help you master this skill:

• Practice, practice, practice: The more problems you solve, the more comfortable you'll become with the process. Try working through various examples with different slopes and points.
• Double-check your work: It's easy to make a small arithmetic error, especially when dealing with fractions and negative signs. Always go back and check your calculations.
• Understand the different forms: Knowing the properties of point-slope and standard form will help you choose the best method for solving a particular problem.

Why is this important?!

Understanding how to convert between different forms of linear equations is a fundamental skill in algebra. It allows you to: interpret the line's properties. Quickly find the x and y intercepts. Solve systems of equations efficiently.

Let's Recap

Let's go over what we've learned, just to make sure it's all clear:

1. Start with Point-Slope: We began with an equation in point-slope form: $y - y_1 = m(x - x_1)$.
2. Distribute the Slope: Multiply the slope ($m$) through the parentheses.
3. Eliminate Fractions: If any are present, multiply the entire equation by the common denominator.
4. Rearrange into Standard Form: Get all the x and y terms on one side and the constant on the other ($Ax + By = C$).
5. Ensure 'A' is Positive: If the coefficient of x is negative, multiply the whole equation by -1.

By following these steps, you can confidently convert any point-slope equation to standard form. Great job, everyone!

Final Thoughts

Converting between forms of linear equations might seem tricky at first, but with practice, it becomes second nature. Remember to focus on each step, keep your signs straight, and don't be afraid to ask for help if you get stuck. Keep up the excellent work, and you'll conquer algebra in no time. Keep practicing, and you'll be acing those math problems in no time. Until next time, keep those equations balanced, and keep up the great work, everyone! You got this! Remember, understanding different forms of linear equations is a key skill, so pat yourselves on the back, you amazing learners!