Slope-Intercept Form: Y + 3 = 4(x - 5) Explained
Hey everyone! Today, we're diving into the fascinating world of linear equations, specifically focusing on how to transform an equation into the slope-intercept form. This form, y = mx + b, is super useful because it immediately tells us the slope (m) and the y-intercept (b) of a line. We're going to tackle the equation y + 3 = 4(x - 5) and walk through the steps to get it into that beautiful y = mx + b format. So, grab your pencils, and let's get started!
Understanding the Slope-Intercept Form
Before we jump into the solution, let's make sure we're all on the same page about what the slope-intercept form actually represents. The equation y = mx + b is a standard way to write a linear equation. Here,
- y represents the vertical coordinate.
- x represents the horizontal coordinate.
- m represents the slope of the line, which tells us how steep the line is and its direction (whether it's going uphill or downhill). It's essentially the "rise over run" – the change in y divided by the change in x.
- b represents the y-intercept, which is the point where the line crosses the y-axis. It's the value of y when x is 0.
Knowing this form is like having a secret decoder ring for lines! When an equation is in slope-intercept form, you can instantly visualize the line's characteristics. You know its steepness and where it crosses the y-axis. This makes it incredibly easy to graph the line, compare it to other lines, and solve related problems.
Now, why is this so important? Well, imagine you're trying to understand a real-world scenario that can be modeled by a line. Maybe you're tracking the growth of a plant over time, or the distance a car travels at a constant speed. If you can express the relationship as a linear equation in slope-intercept form, you can easily interpret the key aspects of the situation. The slope might represent the growth rate of the plant, and the y-intercept could be the plant's initial height. The slope-intercept form translates abstract equations into concrete, understandable information.
Moreover, the slope-intercept form is a fundamental concept in algebra and calculus. It's a building block for understanding more advanced topics like linear transformations, derivatives, and integrals. Mastering this form now will pay dividends as you progress in your mathematical journey. So, let's keep this explanation in mind as we dive into transforming our equation. We're not just rearranging symbols; we're unlocking the secrets of lines!
Step-by-Step Solution: Transforming y + 3 = 4(x - 5) into Slope-Intercept Form
Okay, let's get our hands dirty and transform the equation y + 3 = 4(x - 5) into the slope-intercept form. This is where the magic happens! We'll break it down into manageable steps, making sure each one is crystal clear.
Step 1: Distribute the 4
The first thing we need to do is get rid of those parentheses. We have a 4 lurking outside the (x - 5), and we need to multiply it through. This is called the distributive property, and it's a fundamental tool in algebra. So, we multiply 4 by both x and -5:
- 4 * x = 4x
- 4 * -5 = -20
This transforms our equation into:
- y + 3 = 4x - 20
See? We've already made progress! The parentheses are gone, and the equation is looking a little cleaner. Distributing is a crucial step because it unwraps the expression, allowing us to isolate y later on. It's like peeling back the layers of an onion – we're getting closer to the core!
Step 2: Isolate y
Now, our goal is to get y all by itself on the left side of the equation. To do this, we need to get rid of that pesky +3. Remember, whatever we do to one side of the equation, we must do to the other to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level.
To eliminate the +3, we'll subtract 3 from both sides of the equation:
- y + 3 - 3 = 4x - 20 - 3
On the left side, the +3 and -3 cancel each other out, leaving us with just y. On the right side, we combine -20 and -3 to get -23. This gives us:
- y = 4x - 23
Boom! We've done it! We've successfully isolated y. This is the heart of the whole process. Getting y by itself is what allows us to directly read off the slope and y-intercept.
Step 3: Identify the Slope and y-intercept
Take a good look at our equation: y = 4x - 23. Does it look familiar? It should! It's in the slope-intercept form, y = mx + b. Now, we just need to match up the pieces:
- The number in front of x is the slope, m. In this case, m = 4.
- The constant term (the number without an x) is the y-intercept, b. Here, b = -23.
So, we've discovered that the line has a slope of 4 and crosses the y-axis at the point (0, -23). That's the power of the slope-intercept form – it reveals these key characteristics at a glance.
The Answer and Why It's Correct
Looking back at the options, we can see that the correct answer is:
- D. y = 4x - 23
This is exactly what we arrived at through our step-by-step transformation. It's crucial to understand why this is the right answer. We didn't just blindly follow steps; we understood the underlying principles of the slope-intercept form and how to manipulate equations to achieve it.
Options A, B, and C are incorrect because they don't result from correctly applying the distributive property and isolating y. They might represent common mistakes, like forgetting to distribute to both terms inside the parentheses or adding instead of subtracting when isolating y. This highlights the importance of carefully following each step and double-checking your work.
Common Mistakes to Avoid
Transforming equations can be tricky, and it's easy to make small errors that lead to the wrong answer. Let's talk about some common pitfalls to watch out for:
- Forgetting to Distribute: This is a big one! When you have a number multiplying a set of parentheses, like in our original equation, you must distribute that number to every term inside the parentheses. In our case, we needed to multiply 4 by both x and -5. Failing to do so will throw off your entire calculation.
- Incorrectly Combining Like Terms: Remember, you can only combine terms that have the same variable and exponent (or are constants). You can't combine 4x and -23, for example, because they're not like terms. Mixing these up will lead to an incorrect equation.
- Sign Errors: Signs are sneaky little things! Pay close attention to whether you're adding or subtracting, and make sure you're applying the correct signs when distributing or combining terms. A single sign error can completely change your answer.
- Not Performing the Same Operation on Both Sides: This is the golden rule of equation solving! Whatever you do to one side of the equation, you must do to the other to maintain balance. If you subtract 3 from the left side, you need to subtract 3 from the right side as well. Failing to do this will break the equality.
- Rushing Through the Steps: It's tempting to try and solve equations quickly, but rushing often leads to mistakes. Take your time, write out each step clearly, and double-check your work. It's better to be accurate than fast.
By being aware of these common mistakes, you can actively avoid them and boost your equation-solving skills.
Practice Makes Perfect: More Examples and Exercises
Okay, guys, we've covered a lot of ground here! We've dissected the slope-intercept form, walked through the solution step-by-step, and highlighted common mistakes to avoid. But the real key to mastering this is practice. So, let's look at some more examples and exercises to solidify your understanding.
Example 1
Let's try transforming the equation 2y - 6 = -3x + 4 into slope-intercept form.
- Isolate the term with y: Add 6 to both sides:
- 2y = -3x + 10
- Solve for y: Divide both sides by 2:
- y = (-3/2)x + 5
Now it's in slope-intercept form! The slope is -3/2, and the y-intercept is 5.
Example 2
Let's tackle y - 1 = 5(x + 2).
- Distribute the 5:
- y - 1 = 5x + 10
- Isolate y: Add 1 to both sides:
- y = 5x + 11
Again, we have the slope-intercept form! The slope is 5, and the y-intercept is 11.
Exercises for You to Try
Now, it's your turn! Try transforming these equations into slope-intercept form:
- 3y + 9 = 6x - 3
- -2y + 4 = x + 8
- y + 2 = -4(x - 1)
Work through these exercises carefully, applying the steps we've discussed. Check your answers by graphing the original equation and the transformed equation – they should produce the same line! This is a great way to verify your work and build confidence.
Remember, the more you practice, the more comfortable you'll become with transforming equations. It's like learning any new skill – it takes time and effort, but the rewards are well worth it. So, keep practicing, and you'll be a slope-intercept form pro in no time!
Conclusion: The Power of Slope-Intercept Form
Alright, we've reached the end of our journey into the slope-intercept form! We started by understanding what this form represents, then we tackled the equation y + 3 = 4(x - 5) step-by-step, and finally, we explored some common mistakes and practiced with more examples. Hopefully, you're feeling much more confident about transforming equations and working with the slope-intercept form.
Remember, the slope-intercept form is more than just a mathematical equation; it's a powerful tool for understanding and visualizing linear relationships. It allows us to quickly identify the slope and y-intercept of a line, which in turn helps us graph the line, compare it to other lines, and interpret real-world scenarios modeled by linear equations.
Mastering this form is a crucial step in your mathematical journey. It's a building block for more advanced concepts, and it's a skill that will serve you well in various fields, from science and engineering to economics and finance. So, keep practicing, keep exploring, and never stop learning! You've got this!