Slope-Intercept Form: Rewriting 19x + 5y = 2

Hey guys! Today, we're diving into the world of linear equations and focusing on how to transform one into the slope-intercept form. Specifically, we're going to tackle the equation 19x + 5y = 2. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step, so you'll be a pro in no time. Understanding the slope-intercept form is crucial because it allows us to quickly identify the slope and y-intercept of a line, making graphing and analyzing linear relationships a breeze. So, let’s get started and turn this equation into its slope-intercept best!

Understanding Slope-Intercept Form

Before we jump into the transformation, let's quickly recap what the slope-intercept form actually is. The slope-intercept form is a way of writing linear equations that makes it super easy to identify two key features of a line: its slope and its y-intercept. The general form looks like this:

y = mx + b

Where:

• y is the dependent variable (usually plotted on the vertical axis)
• x is the independent variable (usually plotted on the horizontal axis)
• m is the slope of the line, representing how steeply the line rises or falls. It's the "rise over run," or the change in y for every unit change in x.
• b is the y-intercept, which is the point where the line crosses the y-axis (where x = 0).

Knowing this form is like having a secret decoder ring for linear equations. Once an equation is in slope-intercept form, you can immediately see the slope and y-intercept, which makes graphing the line or comparing it to other lines incredibly simple. So, our mission today is to take the equation 19x + 5y = 2 and rearrange it to fit this y = mx + b format. This involves using algebraic manipulation to isolate y on one side of the equation. Think of it like solving a puzzle where our goal is to get y all by itself!

Step-by-Step Conversion of 19x + 5y = 2

Alright, let's get our hands dirty and transform the equation 19x + 5y = 2 into slope-intercept form. Remember, our goal is to isolate y on one side of the equation. We'll do this step-by-step, so it's crystal clear.

Step 1: Isolate the term with 'y'

The first thing we need to do is get the term with y (which is 5y) by itself on the left side of the equation. To do this, we need to get rid of the 19x term. Since it's being added to 5y, we'll do the opposite operation: subtract 19x from both sides of the equation. This is crucial to keep the equation balanced – whatever you do to one side, you must do to the other.

19x + 5y - 19x = 2 - 19x

This simplifies to:

5y = 2 - 19x

Now we have the 5y term isolated, which is a great start! We're one step closer to getting y all by itself.

Step 2: Divide to isolate 'y'

We're almost there! We have 5y on one side, but we want just y. Currently, y is being multiplied by 5. To undo this multiplication, we'll do the opposite operation: divide both sides of the equation by 5. Again, we need to do this to the entire side of the equation, so every term gets divided by 5.

(5y) / 5 = (2 - 19x) / 5

This simplifies to:

y = (2/5) - (19x/5)

Step 3: Rearrange into slope-intercept form (y = mx + b)

We've got y by itself, which is fantastic! However, to truly be in slope-intercept form, we want the equation in the y = mx + b format. This means the term with x should come before the constant term. It’s a simple rearrangement, but it makes it much easier to identify the slope and y-intercept.

So, we just switch the order of the terms on the right side:

y = -(19/5)x + (2/5)

And there you have it! We've successfully converted the equation 19x + 5y = 2 into slope-intercept form.

Identifying the Slope and Y-intercept

Now that our equation is in the beautiful slope-intercept form (y = -(19/5)x + (2/5)), we can easily identify the slope and y-intercept. Remember, in the form y = mx + b, m represents the slope and b represents the y-intercept.

• Slope (m): Looking at our equation, the coefficient of x is -19/5. This is our slope. It tells us that for every 5 units we move to the right on the graph, the line goes down 19 units. The negative sign indicates that the line is decreasing or sloping downwards from left to right.
• Y-intercept (b): The constant term in our equation is 2/5. This is our y-intercept. It means the line crosses the y-axis at the point (0, 2/5). So, if you were to graph this line, it would intersect the vertical axis at 2/5.

Being able to quickly identify these two key features is the power of the slope-intercept form. It makes graphing lines and understanding their behavior much simpler.

Why is Slope-Intercept Form Useful?

You might be thinking, "Okay, we converted the equation, but why did we even bother?" That’s a great question! The slope-intercept form isn't just a mathematical exercise; it's a powerful tool for understanding and working with linear equations. Here are a few key reasons why it's so useful:

• Easy Graphing: As we've already seen, the slope-intercept form makes graphing lines incredibly easy. You know exactly where the line crosses the y-axis (the y-intercept), and you know how steep the line is and which direction it goes (the slope). You can plot the y-intercept, then use the slope to find another point, and simply draw a line through them.
• Comparing Lines: If you have two or more equations in slope-intercept form, it's very easy to compare their slopes and y-intercepts. You can quickly see if lines are parallel (same slope), perpendicular (slopes are negative reciprocals of each other), or if they intersect. This is invaluable in many applications.
• Modeling Real-World Situations: Linear equations are often used to model real-world relationships, such as the cost of a service based on usage, the distance traveled at a constant speed, or the relationship between temperature and pressure. The slope and y-intercept have real-world meanings in these contexts. For example, the slope might represent the cost per unit of usage, and the y-intercept might represent a fixed starting cost. Having the equation in slope-intercept form helps you easily interpret these values.
• Solving Systems of Equations: The slope-intercept form can be helpful when solving systems of linear equations. You can use substitution or elimination methods more easily when the equations are in this form.

In essence, the slope-intercept form provides a clear and concise way to represent linear relationships, making them easier to analyze, graph, and apply to various situations. It's a fundamental concept in algebra and a skill well worth mastering.

Common Mistakes to Avoid

When converting equations to slope-intercept form, there are a few common pitfalls that students sometimes encounter. Being aware of these mistakes can help you avoid them and ensure you get the correct answer.

• Forgetting to divide all terms: When you divide both sides of the equation to isolate y, remember to divide every term on both sides by the coefficient of y. A common mistake is to divide only the term with y or to forget to divide the constant term. For example, in our problem, we divided both 2 and -19x by 5.
• Incorrectly applying the distributive property: If you have a situation where you need to distribute a number, make sure you do it correctly. This usually doesn't come up directly in this type of problem, but it's a general algebraic skill to keep in mind.
• Not paying attention to signs: Be very careful with positive and negative signs. A simple sign error can throw off your entire answer. Remember the rules for adding, subtracting, multiplying, and dividing with negative numbers.
• Forgetting the order of operations: Always follow the order of operations (PEMDAS/BODMAS). In these types of problems, you typically deal with addition/subtraction before division.
• Not simplifying fractions: If your slope or y-intercept ends up as a fraction, make sure it's simplified to its lowest terms. While an unsimplified fraction isn't technically wrong, it's good mathematical practice to simplify whenever possible.

By being mindful of these common errors and double-checking your work, you can increase your confidence and accuracy in converting equations to slope-intercept form.

Practice Problems

Okay, guys, now that we've gone through the steps and covered some common pitfalls, it's time to put your knowledge to the test! Practice is key to mastering any mathematical concept, and converting equations to slope-intercept form is no exception. Here are a few practice problems for you to try. Grab a pencil and paper, and let's see what you've learned!

1. Rewrite the equation 3x - 2y = 6 in slope-intercept form. Identify the slope and y-intercept.
2. Convert the equation 4y + 8x = 12 to slope-intercept form. What are the slope and y-intercept?
3. Transform the equation -5x + y = -3 into y = mx + b format. State the slope and y-intercept.
4. Rewrite 2x + 3y = 9 in slope-intercept form. What is the slope, and where does the line cross the y-axis?
5. Convert -x - 4y = 8 to slope-intercept form and determine the slope and y-intercept.

Work through these problems step-by-step, using the method we discussed earlier. Remember to isolate the y term, divide by the coefficient of y, and rearrange the equation into the y = mx + b format. Once you've converted the equations, be sure to identify the slope and y-intercept. This will help solidify your understanding of the slope-intercept form.

If you get stuck, don't worry! Go back and review the steps we covered, and pay close attention to the examples. Math is like building a house – each concept builds on the previous one. So, make sure you have a solid foundation, and you'll be able to tackle even more challenging problems in the future.

Conclusion

Alright, guys, we've reached the end of our journey into converting equations to slope-intercept form! We started by understanding what the slope-intercept form (y = mx + b) actually represents and why it's so useful. We then took the equation 19x + 5y = 2 and meticulously transformed it, step-by-step, into its slope-intercept form: y = -(19/5)x + (2/5). Along the way, we identified the slope and y-intercept, discussed the many benefits of using this form, and even highlighted some common mistakes to avoid.

Remember, the key takeaway here is that the slope-intercept form provides a powerful way to visualize and analyze linear relationships. It allows us to quickly identify the slope and y-intercept, making graphing and comparing lines a breeze. Whether you're dealing with abstract mathematical concepts or real-world applications, understanding slope-intercept form is a valuable skill.

So, keep practicing, guys! The more you work with these types of problems, the more comfortable and confident you'll become. And who knows, maybe you'll even start seeing linear equations everywhere you go!