Solve Equations: Convert To Y = Mx + C Form

Hey math enthusiasts! Today, we're diving into a fundamental concept in algebra: rearranging equations into the slope-intercept form, which is written as y = mx + c. This form is super useful because it makes it incredibly easy to identify the slope (m) and the y-intercept (c) of a line. Knowing these two things gives you a complete picture of the line's behavior on a graph – its steepness and where it crosses the y-axis. We will go through the given equations, step by step, showing how to manipulate them to achieve this elegant form. Let's get started, and I promise, by the end of this guide, you'll be a pro at converting equations into the y = mx + c format. Ready, guys? Let's break down each equation, one by one. Understanding this concept is the key to many topics in algebra and beyond, so let's make sure we grasp it. We will solve each equation provided to convert them to y = mx + c.

(a) $2y = 6x + 10$

Alright, let's kick things off with the first equation: 2y = 6x + 10. Our goal here is to isolate y on the left side of the equation. To do this, we need to get rid of the 2 that's multiplying y. The best way to do that is to divide every term in the equation by 2. It’s important to remember that whatever operation you perform on one side of the equation, you must do it on the other side as well to keep things balanced. So, let’s do it step by step, so that everyone can follow: Dividing each term by 2 gives us:

• (2y)/2 = (6x)/2 + 10/2.

Now, let's simplify each part. 2y/2 simplifies to y, 6x/2 simplifies to 3x, and 10/2 simplifies to 5. Putting it all together, we now have:

• y = 3x + 5.

And there you have it! The equation is now in the y = mx + c form, where m (the slope) is 3, and c (the y-intercept) is 5. This tells us the line has a slope of 3 (meaning it goes up 3 units for every 1 unit to the right) and crosses the y-axis at the point (0, 5). Understanding how to manipulate equations like this is like unlocking a secret code in math – it opens up a whole new level of understanding and problem-solving abilities. Don’t worry if it takes a bit of practice. The more you work with these equations, the more comfortable and confident you'll become. So, keep practicing, keep asking questions, and most importantly, keep enjoying the process of learning. The transformation is complete, and the equation now elegantly fits our desired format. From this format, we can easily identify the slope and the y-intercept. This simple step unlocks so much more information about the line.

(b) $3y = 12 - 9x$

Next up, we have 3y = 12 - 9x. This one might look a little different, but the process is the same. We still want to isolate y. So, we’ll start by dividing every term in the equation by 3. This is similar to the last example, but now with some extra terms. Remember, we need to apply the operation to every term to keep everything balanced. Let's see how that looks step by step. Here’s how it breaks down:

• (3y)/3 = 12/3 - (9x)/3.

Now, let's simplify each part. 3y/3 gives us y, 12/3 gives us 4, and (9x)/3 gives us 3x. So, we now have:

• y = 4 - 3x.

We're almost there! To make it look perfectly like y = mx + c, we just need to rearrange the terms a bit. We can rewrite the equation as y = -3x + 4. Now, it's clear that the slope (m) is -3, and the y-intercept (c) is 4. This tells us the line slopes downwards (because the slope is negative) and crosses the y-axis at the point (0, 4). Converting equations to the y = mx + c form isn't just about memorizing steps; it's about developing a solid understanding of how equations work and how to manipulate them to get the information you need. As you practice more, you’ll start to see patterns and shortcuts that make the process even quicker and easier. Remember, every equation is a puzzle, and with a little bit of patience and practice, you can solve it. Keep up the excellent work, and celebrate each small victory. We are getting better at the manipulation of the equations, so that we can easily deduce the slope and y-intercept.

(c) $4x + 2y = 12$

Let’s move on to the equation 4x + 2y = 12. This one has an extra term on the left side, but don’t worry, the process is still straightforward. Our goal is still to get y by itself on one side. The first step is to get rid of the 4x term. We do this by subtracting 4x from both sides of the equation. Always remember to perform the same operation on both sides of the equation to maintain balance.

• 4x + 2y - 4x = 12 - 4x.

This simplifies to 2y = 12 - 4x. Now, just like in our previous examples, we need to isolate y. So, we'll divide every term by 2.

• (2y)/2 = 12/2 - (4x)/2.

Simplifying further, we get y = 6 - 2x. And finally, let’s rearrange the terms to get it into the y = mx + c format, which gives us y = -2x + 6. Now it's in the standard form! Here, the slope (m) is -2, and the y-intercept (c) is 6. This tells us the line slopes downwards and crosses the y-axis at the point (0, 6). By converting to y = mx + c, we gain valuable insights into the line’s properties, allowing us to visualize and analyze it effectively. It's like having a superpower that lets you quickly understand the behavior of any line. Keep in mind that practice is key, and each equation is a chance to refine your skills and boost your confidence. We are getting the hang of it, guys!

(d) $2x + 3y - 7 = 0$

Okay, let's tackle the equation 2x + 3y - 7 = 0. This one might look a bit different because it equals zero, but the process is fundamentally the same. Our mission remains: isolate y. First, we need to get the terms with y on one side of the equation and everything else on the other side. This time we have a subtraction, so we need to add to both sides. So, let’s start by adding 7 to both sides and subtracting 2x from both sides. We perform these operations to move all the terms that do not include the y term to the other side of the equation.

• 2x + 3y - 7 + 7 - 2x = 0 + 7 - 2x.

This simplifies to 3y = -2x + 7. Now, just like before, we divide every term by 3 to isolate y.

• (3y)/3 = (-2x)/3 + 7/3.

This simplifies to y = -2/3x + 7/3. And there we have it! It's in the y = mx + c form, where the slope (m) is -2/3, and the y-intercept (c) is 7/3. This means the line slopes downwards at a less steep angle, and it crosses the y-axis at the point (0, 7/3). The ability to convert equations into the y = mx + c form is a fundamental skill in algebra. The more you work with these equations, the more comfortable and confident you will become. Remember, every equation is a puzzle, and with a little bit of patience and practice, you can solve it. You're doing great; keep up the momentum!

(e) $9x - 3y = 21$

Let’s take on 9x - 3y = 21. Similar to our previous examples, we aim to isolate y. The first step here is to move the 9x term to the other side. So, we'll subtract 9x from both sides of the equation. This gets the equation one step closer to isolating the y term. Remember, we always perform the same operation on both sides to keep the balance.

• 9x - 3y - 9x = 21 - 9x.

Which simplifies to -3y = -9x + 21. Now, to get y by itself, we divide every term by -3.

• (-3y)/-3 = (-9x)/-3 + 21/-3.

Simplifying further, we get y = 3x - 7. Now we have our equation in the y = mx + c form. Here, the slope (m) is 3, and the y-intercept (c) is -7. This tells us the line slopes upwards and crosses the y-axis at the point (0, -7). By mastering these techniques, you are building a strong foundation for more advanced math concepts. Each step you take is a step closer to understanding and mastering the principles of algebra. So, keep up the fantastic work and embrace the process of learning.

(f) $2x - 5y - 8 = 0$

Alright, let’s finish strong with 2x - 5y - 8 = 0. Just like with equation (d), we'll start by isolating the y term. First, let's add 8 to both sides and subtract 2x from both sides. This shifts all the constants and x terms to the other side of the equation.

• 2x - 5y - 8 + 8 - 2x = 0 + 8 - 2x.

Which simplifies to -5y = -2x + 8. Next, we'll divide every term by -5 to isolate y.

• (-5y)/-5 = (-2x)/-5 + 8/-5.

This simplifies to y = 2/5x - 8/5. And there we have it, our final answer! The equation is now in the y = mx + c form. Here, the slope (m) is 2/5, and the y-intercept (c) is -8/5. This means the line slopes upwards at a gentle angle and crosses the y-axis at the point (0, -8/5). Congratulations, guys! You've successfully rearranged all the equations into the y = mx + c form. This skill is a cornerstone of algebra, and with practice, you'll become more and more proficient. Keep exploring, keep practicing, and remember that every problem you solve is a victory. Keep up the awesome work, you've got this!