Hey guys! Today, we are diving deep into the world of linear equations, specifically focusing on how to find the slope and y-intercept of a given equation. We've got a fun problem to tackle: What is the slope and y-intercept of the equation 3(y - 2) + 6(x + 1) - 2 = 0? This is a classic algebra question, and mastering it will seriously boost your math skills. Let’s break it down step by step so you can confidently solve similar problems in the future. We will explore in detail how to manipulate the equation, identify the slope and y-intercept, and understand the significance of these values in the context of a linear equation.
Understanding Slope and Y-Intercept
Before we jump into solving the problem, let's quickly recap what slope and y-intercept actually mean. These two concepts are fundamental to understanding linear equations and their graphical representation. Think of them as the key ingredients that define a line on a graph. The slope tells us how steep the line is and in which direction it's going – whether it's climbing uphill or sliding downhill. The y-intercept, on the other hand, is the point where the line crosses the y-axis. It's like the line's starting point on the vertical axis. Let's delve deeper into each of these concepts.
Slope: The Steepness of the Line
The slope is a measure of the steepness and direction of a line. It tells us how much the y-value changes for every unit change in the x-value. In simpler terms, it’s the “rise over run.” A positive slope means the line is going upwards as you move from left to right, while a negative slope means the line is going downwards. A larger absolute value of the slope indicates a steeper line, whereas a slope of zero means the line is horizontal. To calculate the slope, we often use the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line. However, the easiest way to identify the slope is when the equation is in slope-intercept form, which we’ll discuss shortly. Understanding the slope is crucial because it gives us immediate insight into the behavior of the linear relationship being represented. Imagine skiing down a hill; the slope is what determines how thrilling (or terrifying!) the ride will be. Similarly, in real-world applications, slope can represent rates of change, such as the speed of a car or the growth rate of a population.
Y-Intercept: Where the Line Crosses the Y-Axis
The y-intercept is the point where the line intersects the y-axis. This is the point where the x-value is zero. The y-intercept is usually denoted as the point (0, b), where b is the y-coordinate. This value is significant because it represents the starting value of the function when x is zero. Think of it as the initial condition or the baseline value. In the context of a graph, it’s super easy to spot the y-intercept – just look for where the line crosses the vertical axis. The y-intercept is particularly useful in various applications. For instance, in a cost equation, the y-intercept might represent the fixed costs, while in a savings scenario, it could represent the initial amount saved. Recognizing the y-intercept helps us quickly grasp the starting point of a linear relationship and provides a crucial reference point for understanding the entire graph.
Transforming the Equation into Slope-Intercept Form
The key to finding the slope and y-intercept in our equation is to transform it into the slope-intercept form. This form is like the secret code that unlocks the values we need. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. Our given equation, 3(y - 2) + 6(x + 1) - 2 = 0, looks a bit messy right now, but don't worry! We're going to clean it up using some basic algebraic techniques. The goal is to isolate y on one side of the equation so that we can clearly see the values of m and b. Let's get started with the transformation process.
Step 1: Distribute the Constants
First, we need to distribute the constants outside the parentheses. This means multiplying the 3 by both terms inside the first set of parentheses (y - 2) and multiplying the 6 by both terms inside the second set of parentheses (x + 1). This step helps us to remove the parentheses and simplify the equation. So, let’s do the math:
3(y - 2) becomes 3y - 6
6(x + 1) becomes 6x + 6
Now, let's substitute these back into our original equation: 3y - 6 + 6x + 6 - 2 = 0. See how much simpler it's already looking? Distributing the constants is a crucial first step because it allows us to combine like terms and move closer to isolating y. It’s like unwrapping a gift – we’re revealing the simpler form hidden beneath the initial complexity.
Step 2: Combine Like Terms
Next up, we need to combine any like terms in the equation. Like terms are terms that have the same variable raised to the same power. In our equation, 3y - 6 + 6x + 6 - 2 = 0, we can see that -6 and +6 are like terms (they are both constants). Combining them is straightforward: -6 + 6 = 0. So, these terms cancel each other out! This leaves us with a much cleaner equation: 3y + 6x - 2 = 0. Combining like terms is a fundamental algebraic skill that simplifies equations and makes them easier to work with. It’s like decluttering a room – by grouping similar items, you create order and clarity.
Step 3: Isolate the Term with y
Now, our goal is to isolate the term with y, which is 3y. To do this, we need to move all other terms to the other side of the equation. We have 3y + 6x - 2 = 0. Let’s start by adding 2 to both sides of the equation to get rid of the -2: 3y + 6x = 2. Next, we need to get rid of the 6x term. To do this, we subtract 6x from both sides: 3y = -6x + 2. We’re getting closer! Isolating the y term is like setting the stage for the final act – we're preparing the equation to reveal its slope-intercept form.
Step 4: Solve for y
The final step in transforming our equation is to solve for y. Currently, we have 3y = -6x + 2. To get y by itself, we need to divide every term in the equation by 3. So, we divide both sides by 3: (3y/3) = (-6x/3) + (2/3). This simplifies to y = -2x + 2/3. Ta-da! We’ve successfully transformed our equation into slope-intercept form. Solving for y is like the grand finale of our algebraic performance – it’s the moment when everything comes together, and the slope and y-intercept are revealed in their full glory.
Identifying the Slope and Y-Intercept
Now that we have our equation in slope-intercept form, y = -2x + 2/3, identifying the slope and y-intercept is a piece of cake! Remember, the slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. By comparing our equation with the slope-intercept form, we can easily pick out the values we need. It’s like reading a map – once you know the legend, you can easily understand the symbols and find your way. Let’s pinpoint the slope and y-intercept in our equation.
Determining the Slope
The slope, m, is the coefficient of the x term in the slope-intercept form. In our equation, y = -2x + 2/3, the coefficient of x is -2. Therefore, the slope of the line is -2. This tells us that the line is decreasing (going downwards) as we move from left to right on the graph. For every 1 unit we move to the right, the y-value decreases by 2 units. A negative slope is like skiing downhill – the higher the absolute value, the steeper the descent. Understanding the slope helps us visualize the direction and steepness of the line.
Determining the Y-Intercept
The y-intercept, b, is the constant term in the slope-intercept form. In our equation, y = -2x + 2/3, the constant term is 2/3. This means the line intersects the y-axis at the point (0, 2/3). The y-intercept is the starting point of the line on the y-axis. It’s like the initial investment in a savings account – it’s the amount you start with before any interest accrues. Knowing the y-intercept gives us a crucial reference point for understanding the linear relationship.
The Solution
So, after all our hard work, we’ve found that the slope of the equation 3(y - 2) + 6(x + 1) - 2 = 0 is -2, and the y-intercept is 2/3. This corresponds to option A: slope = -2, y-intercept = 2/3. Yay, we did it! Understanding how to find the slope and y-intercept is a fundamental skill in algebra, and you’ve just mastered it. Remember, the slope tells us about the steepness and direction of the line, while the y-intercept tells us where the line crosses the y-axis. These two values give us a complete picture of the linear equation. Keep practicing, and you’ll become a pro at solving these types of problems.
Why This Matters
Understanding slope and y-intercept isn't just about solving equations; it’s about understanding relationships. These concepts show up everywhere in real life, from calculating rates of change to predicting trends. For example, if you're tracking your savings, the slope could represent how much you save each month, and the y-intercept could be your initial savings. Or, if you're looking at a graph of a car's speed over time, the slope could represent the car's acceleration, and the y-intercept could be the car's starting speed. Linear equations are a powerful tool for modeling real-world situations, and mastering the slope and y-intercept is key to unlocking that power. So, whether you're solving math problems or analyzing real-world data, these concepts will serve you well.
What are the slope and y-intercept of the equation 3(y - 2) + 6(x + 1) - 2 = 0?
