Solving -6x + Y = 8: Find Slope & Y-Intercept
Let's dive into the world of linear equations! In this article, we'll break down how to solve the linear equation -6x + y = 8 for y, transforming it into the ever-so-useful slope-intercept form. From there, we’ll easily identify the slope and y-intercept. This is a fundamental concept in algebra, and mastering it will set you up for success in more advanced mathematical topics. So, grab your pencil and paper, and let's get started!
Understanding Slope-Intercept Form
Before we tackle the equation itself, it's essential to understand 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 and b represents the y-intercept. The slope (m) tells us how steep the line is and whether it's increasing or decreasing. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. The y-intercept (b) is the point where the line crosses the y-axis. This is the value of y when x is equal to 0. Understanding this form is crucial because it allows us to quickly visualize and analyze linear relationships.
Knowing the slope and y-intercept makes graphing a line incredibly straightforward. You can plot the y-intercept on the graph, and then use the slope to find another point. For example, if the slope is 2 (which can be written as 2/1), you can move 1 unit to the right from the y-intercept and then 2 units up to find another point. Connect the dots, and you've got your line! Slope-intercept form isn't just about graphing, though. It's a powerful tool for understanding how variables relate to each other. In real-world scenarios, the slope might represent a rate of change, like the speed of a car or the cost per item, while the y-intercept could represent a starting value or a fixed cost. By understanding these components, you can interpret and make predictions about various situations.
Step-by-Step Solution: Solving for y
Our mission is to transform the equation -6x + y = 8 into slope-intercept form. Remember, this means isolating y on one side of the equation. To do this, we need to get rid of the -6x term on the left side. The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other. So, to eliminate the -6x, we'll add 6x to both sides of the equation. This might seem like a simple step, but it's the foundation for solving many algebraic problems.
Here's how it looks:
-6x + y + 6x = 8 + 6x
Notice that on the left side, -6x and +6x cancel each other out, leaving us with just y. On the right side, we have 8 + 6x. Now, to make it perfectly align with the slope-intercept form (y = mx + b), we simply rearrange the terms on the right side to put the x term first. This is a matter of convention and makes it easier to identify the slope and y-intercept at a glance. So, we rewrite 8 + 6x as 6x + 8. And just like that, we've solved for y! Our equation in slope-intercept form is:
y = 6x + 8
Identifying the Slope and Y-Intercept
Now that our equation is in slope-intercept form (y = 6x + 8), identifying the slope and y-intercept is a piece of cake. Remember, the slope is the coefficient of x, which is the number multiplying x, and the y-intercept is the constant term, which is the number that's added or subtracted. In our equation, the coefficient of x is 6, and the constant term is 8. Therefore, the slope of the line is 6, and the y-intercept is 8. It's as simple as that!
The slope of 6 tells us that for every 1 unit we move to the right on the graph, the line goes up 6 units. This indicates a steep, upward-sloping line. The y-intercept of 8 tells us that the line crosses the y-axis at the point (0, 8). This is our starting point when graphing the line. Understanding the slope and y-intercept gives us a complete picture of the line's behavior. We know its steepness, its direction, and where it intersects the y-axis. This information is incredibly valuable for graphing, analyzing, and applying linear equations in various contexts. You can now visualize this line in your mind – a steep line that crosses the y-axis at 8. This visual understanding is key to mastering linear equations.
Graphing the Line
Now that we know the slope and y-intercept, let's visualize our equation by graphing the line. Graphing helps solidify our understanding and provides a visual representation of the linear relationship. To graph the line, we'll use the slope and y-intercept we just identified: a slope of 6 and a y-intercept of 8. The first step is to plot the y-intercept, which is the point (0, 8), on the coordinate plane. This is where the line crosses the y-axis. It's our anchor point for drawing the line.
Next, we'll use the slope to find another point on the line. Remember, the slope is 6, which can be thought of as 6/1. This means that for every 1 unit we move to the right along the x-axis, we move 6 units up along the y-axis. Starting from the y-intercept (0, 8), we move 1 unit to the right and 6 units up. This brings us to the point (1, 14). Now we have two points: (0, 8) and (1, 14). With two points, we can draw a straight line that passes through both of them. Use a ruler or straightedge to connect the points, extending the line in both directions as far as you can. This line represents the equation y = 6x + 8. By graphing the line, we can visually confirm our calculations and gain a deeper understanding of the equation. The steepness of the line reflects the slope of 6, and the point where it crosses the y-axis confirms our y-intercept of 8. Graphing is a powerful tool for visualizing and understanding linear equations.
Real-World Applications
Linear equations aren't just abstract mathematical concepts; they have countless real-world applications. Understanding how to solve them and interpret their slopes and y-intercepts can be incredibly useful in everyday life and various professions. For example, imagine you're planning a road trip. The equation y = 60x + 100 could represent the total distance (y) you'll travel after x hours, assuming you're driving at an average speed of 60 miles per hour and you started 100 miles from your destination. In this case, the slope (60) represents your speed, and the y-intercept (100) represents your initial distance from the destination. By understanding this equation, you can predict how far you'll travel in a given amount of time.
Another common application is in budgeting and finance. Let's say you're saving money for a new gadget. The equation y = 50x + 200 could represent your total savings (y) after x weeks, if you save $50 per week and you already have $200 saved. Here, the slope (50) is your weekly savings, and the y-intercept (200) is your initial savings. This equation helps you track your progress and determine how long it will take to reach your savings goal. Linear equations are also used in science and engineering to model relationships between variables, such as temperature and pressure, or force and distance. In business, they can be used to analyze costs, revenue, and profit. The ability to solve and interpret linear equations is a valuable skill that can help you make informed decisions in various aspects of life.
Conclusion
We've successfully navigated the linear equation -6x + y = 8, solved for y, identified the slope and y-intercept, and even explored real-world applications. By converting the equation to slope-intercept form (y = 6x + 8), we easily found that the slope is 6 and the y-intercept is 8. These fundamental concepts are the building blocks for more advanced algebraic topics. Mastering them will give you a solid foundation for tackling complex problems and understanding the relationships between variables.
Remember, solving for y and identifying the slope and y-intercept are powerful tools for analyzing and visualizing linear equations. Whether you're graphing a line, predicting travel distances, or managing your finances, these skills will serve you well. Keep practicing, and you'll become a linear equation pro in no time! For further exploration and practice, consider checking out resources like Khan Academy's Algebra 1 section on linear equations. Happy solving!
