Inverse Of Linear Equation: Step-by-Step Solution

Hey guys! Today, we're diving into a fundamental concept in algebra: finding the inverse of a linear equation. Specifically, we're going to tackle the equation ${ y - 2 = \frac{3}{4}(x + 5) }$ and walk through the process of determining its inverse. Not only will we show all the steps involved, but we'll also express our final answer in the familiar slope-intercept form, ${ y = mx + b }$. So, grab your pencils and notebooks, and let's get started!

Understanding Inverse Functions

Before we jump into the nitty-gritty details of our specific equation, let's take a moment to understand what an inverse function actually is. Think of a function like a machine: you feed it an input (let's call it ${ x }$), and it spits out an output (which we often call ${ y }$). The inverse function is like a machine that undoes what the original function did. If you feed the output ${ y }$ of the original function into its inverse, you'll get back the original input ${ x }$. Inverse functions are a crucial concept in mathematics, allowing us to reverse processes and solve equations in new and interesting ways. Understanding them opens doors to more complex mathematical concepts, and it all starts with mastering the basics.

In simpler terms, if a function takes ${ x }$ to ${ y }$, then the inverse function takes ${ y }$ back to ${ x }$. Graphically, this means that the graphs of a function and its inverse are reflections of each other across the line ${ y = x }$. This visual representation can be a helpful tool in verifying that you've found the correct inverse. Remember, not every function has an inverse! For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output and vice versa. Linear equations (except for horizontal lines) are one-to-one, so we can confidently find the inverse of our given equation.

Step-by-Step Solution: Finding the Inverse

Now, let's get down to business and find the inverse of our equation: ${ y - 2 = \frac{3}{4}(x + 5) }$. We'll break this down into clear, manageable steps so you can follow along easily. Remember, the key to finding the inverse is to swap the roles of ${ x }$ and ${ y }$ and then solve for ${ y }$.

Step 1: Swap ${ x }$ and ${ y }$

This is the fundamental step in finding the inverse. We simply replace every ${ y }$ with an ${ x }$ and every ${ x }$ with a ${ y }$. So, our equation ${ y - 2 = \frac{3}{4}(x + 5) }$ becomes:

${ x - 2 = \frac{3}{4}(y + 5) }$

This swap is the heart of the inverse function concept. By interchanging ${ x }$ and ${ y }$, we're essentially reversing the input and output roles, setting the stage for solving for the new ${ y }$, which will represent our inverse function.

Step 2: Isolate the term with ${ y }$

Our next goal is to isolate the term containing ${ y }$, which is ${ \frac{3}{4}(y + 5) }$. To do this, we'll work to get this term by itself on one side of the equation. Currently, we have ${ x - 2 }$ on the left side. This step is crucial because it brings us closer to solving for ${ y }$ and expressing the inverse function explicitly. It's like peeling away the layers of the equation to reveal the variable we're interested in.

To start, we can multiply both sides of the equation by ${ \frac{4}{3} }$, which is the reciprocal of ${ \frac{3}{4} }$. This will cancel out the fraction on the right side:

${ \frac{4}{3}(x - 2) = \frac{4}{3} \cdot \frac{3}{4}(y + 5) }$

Simplifying this, we get:

${ \frac{4}{3}(x - 2) = y + 5 }$

Step 3: Isolate ${ y }$

Now we're getting closer! We have ${ \frac{4}{3}(x - 2) = y + 5 }$. To completely isolate ${ y }$, we need to get rid of the ${ +5 }$ on the right side. We can do this by subtracting 5 from both sides of the equation:

${ \frac{4}{3}(x - 2) - 5 = y + 5 - 5 }$

This simplifies to:

${ \frac{4}{3}(x - 2) - 5 = y }$

Step 4: Simplify and Write in Slope-Intercept Form

We've successfully isolated ${ y }$, but our job isn't quite done yet. The problem asks us to write our answer in slope-intercept form, which is ${ y = mx + b }$, where ${ m }$ is the slope and ${ b }$ is the y-intercept. Slope-intercept form is a powerful way to represent linear equations because it directly reveals the slope and y-intercept, making it easy to graph and analyze the line.

Let's simplify the left side of our equation and rearrange it to fit this form. First, we'll distribute the ${ \frac{4}{3} }$:

${ y = \frac{4}{3}x - \frac{4}{3}(2) - 5 }$

${ y = \frac{4}{3}x - \frac{8}{3} - 5 }$

Next, we need to combine the constant terms. To do this, we'll express 5 as a fraction with a denominator of 3:

${ y = \frac{4}{3}x - \frac{8}{3} - \frac{15}{3} }$

Now we can combine the fractions:

${ y = \frac{4}{3}x - \frac{23}{3} }$

The Inverse in Slope-Intercept Form

And there you have it! We've successfully found the inverse of the given equation and written it in slope-intercept form. Our final answer is:

${ y = \frac{4}{3}x - \frac{23}{3} }$

This equation represents the inverse of the original equation, ${ y - 2 = \frac{3}{4}(x + 5) }$. This final equation clearly shows the slope (${ \frac{4}{3} }$) and the y-intercept (${ -\frac{23}{3} }$) of the inverse function. You can now easily graph this line or use its equation to solve related problems.

Verification (Optional but Recommended)

To be absolutely sure we've got the correct inverse, we can perform a quick verification. Remember that if two functions are inverses of each other, then composing them (i.e., plugging one into the other) should result in just ${ x }$. While we won't go through the entire verification process here, it's a valuable technique to keep in mind when working with inverse functions. Verification is a crucial step in mathematics to ensure the accuracy of your results. It adds a layer of confidence to your solution and helps you catch any potential errors.

Key Takeaways

Let's recap the key steps we took to find the inverse of our linear equation:

1. Swap ${ x }$ and ${ y }$: This is the fundamental step in finding the inverse.
2. Isolate the term with ${ y }$: Manipulate the equation to get the term containing ${ y }$ by itself.
3. Isolate ${ y }$: Solve for ${ y }$ to express the inverse function explicitly.
4. Simplify and write in slope-intercept form: Put your answer in the desired format, ${ y = mx + b }$, if required.

Finding the inverse of a function might seem tricky at first, but with practice, you'll become a pro! The key is to follow these steps methodically and remember the core concept of reversing the input and output. Mastering these steps will not only help you solve similar problems but also build a solid foundation for more advanced mathematical concepts.

Practice Makes Perfect

Now that we've worked through this example together, the best way to solidify your understanding is to practice! Try finding the inverses of other linear equations. You can even try this method with other types of functions, although the process might be slightly different. Keep practicing, and you'll be solving inverse function problems like a champ in no time! Consistent practice is the key to mastering any mathematical concept. The more you practice, the more comfortable and confident you'll become.

Conclusion

Finding the inverse of a linear equation is a valuable skill in algebra. By following the steps we've outlined, you can confidently tackle these problems and express your answers in slope-intercept form. Remember to swap ${ x }$ and ${ y }$, isolate ${ y }$, and simplify. Keep practicing, and you'll master this concept in no time. Happy solving, guys!