Equation Of A Line: Y-Intercept (0,6) & Slope 3
Hey guys! Today, let's dive into a common problem in algebra: finding the equation of a line when we know its y-intercept and slope. This is super useful in various real-world applications, from calculating distances to predicting trends. So, let's break it down step by step and make sure we understand the concept inside and out.
Understanding Slope-Intercept Form
Before we jump into solving our specific problem, it’s essential to grasp the slope-intercept form of a linear equation. This form is expressed as:
y = mx + b
Where:
yis the dependent variable (usually plotted on the vertical axis).xis the independent variable (usually plotted on the horizontal axis).mis the slope of the line, indicating its steepness and direction.bis the y-intercept, which is the point where the line crosses the y-axis (when x = 0).
This form is incredibly handy because it directly tells us two crucial pieces of information about the line: its slope and where it intersects the y-axis. Knowing these two things makes it super easy to write the equation of the line. The slope, often denoted as m, tells us how much the y value changes for every unit change in the x value. A positive slope means the line goes upwards as you move from left to right, while a negative slope means the line goes downwards. The steeper the line, the larger the absolute value of the slope. The y-intercept, denoted as b, is the point where the line crosses the y-axis. This is the value of y when x is 0. Think of it as the starting point of the line on the vertical axis. Understanding these components is crucial for anyone delving into linear equations. The slope-intercept form allows us to quickly visualize and analyze linear relationships. For instance, if we know the slope is 2, we know that for every 1 unit increase in x, y increases by 2 units. If the y-intercept is 3, we know the line crosses the y-axis at the point (0, 3). This form is not just a mathematical formula; it’s a powerful tool for understanding and predicting linear behavior in various scenarios. From graphing lines to solving real-world problems, the slope-intercept form provides a clear and concise way to represent linear equations. Let's keep this concept in mind as we move forward and tackle some examples. By understanding slope and y-intercept, we can easily construct and interpret linear equations.
Our Problem: Y-Intercept (0,6) and Slope 3
Now, let’s tackle the specific problem we have. We’re given that the y-intercept is (0, 6) and the slope is 3. This means:
b = 6(since the y-intercept is the y-value when x = 0)m = 3(the slope is given as 3)
With this information, we can directly plug these values into the slope-intercept form (y = mx + b). So, we have m which represents the slope, and b which is the y-intercept. The slope, in this case, is 3, meaning for every unit increase in x, the y value increases by 3 units. The y-intercept is 6, meaning the line crosses the y-axis at the point (0, 6). Armed with this information, we're ready to put it all together into the slope-intercept form of the equation. This is where the magic happens! We're taking two key pieces of information about the line and combining them to create a complete equation that describes its behavior. So, let's get to it and write out the equation step by step. This is a fundamental skill in algebra, and mastering it will open up a whole new world of problem-solving possibilities. It's like having a secret code to unlock the mysteries of linear relationships. By understanding how the slope and y-intercept work together, we can confidently tackle any linear equation problem that comes our way. So, keep practicing and refining your skills. The more you work with these concepts, the more intuitive they will become. And remember, math isn't about memorizing formulas; it's about understanding the underlying principles. Once you grasp the concepts, the formulas will naturally fall into place. So, let's dive in and see how we can apply this knowledge to our specific problem. Remember, we're just plugging in the given values into the slope-intercept form, so it's a pretty straightforward process.
Plugging in the Values
Substitute m = 3 and b = 6 into the equation y = mx + b:
y = (3)x + (6)
Simplify it:
y = 3x + 6
And that’s it! We've found the equation of the line. This equation, y = 3x + 6, completely describes the line with a slope of 3 and a y-intercept of (0, 6). It tells us everything we need to know about the line’s behavior on the coordinate plane. This is the final equation that represents the line we've been working with. It's a simple yet powerful expression that captures the essence of the line's characteristics. But what does this equation actually mean? Well, it means that for any point (x, y) on this line, the y-coordinate is equal to 3 times the x-coordinate, plus 6. So, if you give me any x-value, I can plug it into this equation and find the corresponding y-value that lies on the line. Isn't that cool? It's like having a magic formula that can generate all the points on the line. But more than that, this equation allows us to analyze and predict the behavior of the line. We can see that as x increases, y increases at a rate of 3, thanks to the slope. We also know that the line crosses the y-axis at the point (0, 6), which is our y-intercept. These pieces of information give us a complete picture of the line's position and direction on the graph. So, next time you see an equation like this, remember that it's not just a bunch of symbols; it's a story about a line and its unique characteristics. By understanding the slope-intercept form, we can decode these stories and gain valuable insights into the world of linear relationships. This skill is not only essential for math class but also for many real-world applications, such as analyzing data, predicting trends, and making informed decisions.
Final Answer
The equation of the line with a y-intercept of (0, 6) and a slope of 3 is:
y = 3x + 6
So, there you have it! Finding the equation of a line given its slope and y-intercept is a straightforward process once you understand the slope-intercept form. Just plug in the values and simplify. Remember this equation; it's a fundamental concept in algebra and will be super useful in your math journey.