Slope & Y-intercept: 0.5x + 0.25y = 0.75 In Slope-Intercept Form
Hey guys! Let's dive into some math and tackle a common problem: finding the slope and y-intercept of a linear equation. Specifically, we're going to work with the equation 0.5x + 0.25y = 0.75. The ultimate goal? To express this equation in the familiar slope-intercept form. This is a fundamental concept in algebra, and mastering it will help you understand and visualize linear relationships much better. So, grab your pencils, and let's get started!
What is Slope-Intercept Form?
Before we jump into the nitty-gritty, let's quickly recap what slope-intercept form actually is. The slope-intercept form of a linear equation is expressed as y = mx + b, where:
- m represents the slope of the line.
- b represents the y-intercept (the point where the line crosses the y-axis).
This form is super useful because it tells us two key things about the line at a glance: its steepness (slope) and where it intersects the vertical axis (y-intercept). By converting our equation to this form, we can easily identify these values. The slope, often called the rate of change, indicates how much y changes for every unit change in x. The y-intercept gives us a starting point on the graph, helping us visualize the line's position on the coordinate plane. This understanding is crucial in various real-world applications, from calculating rates and distances to understanding financial trends and modeling physical phenomena. So, let's move on to the first step in our process: isolating y on one side of the equation.
Step 1: Isolate the 'y' Term
Our first mission is to isolate the y term in the equation 0.5x + 0.25y = 0.75. To do this, we need to get rid of the 0.5x term on the left side. How do we do that? We use the power of inverse operations! We'll subtract 0.5x from both sides of the equation. This maintains the equality and starts us on the path to getting y by itself. Think of it like balancing a scale: whatever we do to one side, we must do to the other. So, let's perform the subtraction:
0. 5x + 0.25y - 0.5x = 0.75 - 0.5x
This simplifies to:
0. 25y = 0.75 - 0.5x
Great! We're one step closer. Now the y term is partially isolated. We have 0.25y on one side, but we want just y. What's our next move? We need to get rid of that 0.25 coefficient. This leads us to the next step, where we'll divide both sides of the equation by 0.25. This will completely isolate y and bring us closer to the slope-intercept form we're aiming for. So, let's head on to step two and finish the job!
Step 2: Divide to Isolate 'y'
Now that we have 0.25y = 0.75 - 0.5x, we need to isolate y completely. To do this, we'll divide both sides of the equation by 0.25, which is the coefficient of y. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. So, let's perform the division:
(0.25y) / 0.25 = (0.75 - 0.5x) / 0.25
This simplifies to:
y = (0.75 / 0.25) - (0.5x / 0.25)
Now, let's do the math! 0.75 / 0.25 equals 3, and 0.5 / 0.25 equals 2. So, our equation becomes:
y = 3 - 2x
We're almost there! We have y isolated, but the equation is not quite in the standard slope-intercept form (y = mx + b) yet. The x term is on the wrong side. No worries, we can easily fix that. In the next step, we'll simply rearrange the terms to get our equation into the familiar y = mx + b format. This will make it super clear what the slope and y-intercept are. So, let's move on to the final step and put the finishing touches on our equation!
Step 3: Rewrite in Slope-Intercept Form (y = mx + b)
Okay, we've got y = 3 - 2x. It's close, but not quite the y = mx + b form we're after. The only thing we need to do is rearrange the terms so that the x term comes first. Remember, the order of addition doesn't change the value, so we can simply swap the positions of 3 and -2x. This gives us:
y = -2x + 3
Bingo! We've successfully converted our original equation into slope-intercept form. Now, it's super easy to identify the slope and y-intercept. By comparing our equation y = -2x + 3 to the standard form y = mx + b, we can see that:
- The slope (m) is -2.
- The y-intercept (b) is 3.
That's it! We've found the slope and y-intercept. This means our line has a negative slope, so it goes downwards as we move from left to right. For every one unit we move to the right on the graph, the line goes down two units. The y-intercept of 3 tells us that the line crosses the y-axis at the point (0, 3). This completes our task. Let's recap the whole process to make sure we've got it down.
Conclusion: Slope is -2 and Y-intercept is 3
So, let's recap what we've done. We started with the equation 0.5x + 0.25y = 0.75 and our mission was to find the slope and y-intercept by converting it to slope-intercept form (y = mx + b). Here's a quick rundown of the steps:
- Isolate the 'y' term: We subtracted 0.5x from both sides, resulting in 0.25y = 0.75 - 0.5x.
- Divide to isolate 'y': We divided both sides by 0.25, which gave us y = 3 - 2x.
- Rewrite in slope-intercept form: We rearranged the terms to get y = -2x + 3.
From this final equation, we easily identified that the slope is -2 and the y-intercept is 3. You see, by understanding the principles of algebra and using inverse operations, we can manipulate equations to reveal valuable information. This skill is not just useful for math class; it's a powerful tool for problem-solving in many areas of life. Whether you're calculating distances, analyzing data, or even just figuring out the best deal at the store, understanding linear equations and their components like slope and y-intercept can give you a real edge. So, keep practicing, keep exploring, and you'll become a math whiz in no time!