Find The Line Equation Slope -2/3 And Y-intercept (0,4) Explained

by James Vasile 66 views

Hey guys! Let's dive into the fascinating world of linear equations and explore how to pinpoint the equation of a line given its slope and y-intercept. Today, we're tackling a specific problem: finding the equation of a line with a slope of -2/3 that gracefully crosses the y-axis at the point (0,4). Buckle up, because we're about to unravel this mathematical mystery!

Understanding the Slope-Intercept Form

Before we jump into solving the problem, it's crucial to grasp the slope-intercept form of a linear equation. This form, represented as y = mx + b, is our trusty key to unlocking the equation we seek. Let's break it down:

  • y: This represents the vertical coordinate on the Cartesian plane.
  • m: Ah, the slope! This value tells us how steep the line is and whether it's ascending or descending. A negative slope, like ours (-2/3), indicates a line that slopes downward from left to right.
  • x: This represents the horizontal coordinate on the Cartesian plane.
  • b: This is the y-intercept, the point where the line intersects the y-axis. In our case, this is the point (0,4), meaning b = 4.

Deciphering the Given Information

Now that we're fluent in the language of slope-intercept form, let's dissect the information we've been given. We know two crucial pieces of the puzzle:

  1. The slope (m) of the line is -2/3. This means for every 3 units we move to the right along the x-axis, the line descends 2 units along the y-axis.
  2. The line crosses the y-axis at the point (0,4). This tells us that the y-intercept (b) is 4.

Plugging in the Values

With our newfound knowledge, we can now plug these values directly into the slope-intercept form:

y = mx + b

Substitute m with -2/3 and b with 4:

y = (-2/3)x + 4

And there you have it! We've successfully crafted the equation of the line.

Why Other Options Are Incorrect

Let's take a moment to examine why the other options provided are not the correct answer. This will solidify our understanding and prevent future missteps.

  • Option A: y = 4x - 2/3

    This equation has the correct form (y = mx + b), but it incorrectly assigns the slope and y-intercept. The slope is 4, and the y-intercept is -2/3, which contradicts the given information.

  • Option B: x = 4y - 2/3

    This equation is not in slope-intercept form. While it represents a linear relationship, it's solved for x rather than y. To get it into slope-intercept form, we'd need to rearrange it, and it wouldn't match our given conditions.

  • Option C: x = (-2/3)y + 4

    Similar to option B, this equation is not in slope-intercept form. It's solved for x, and rearranging it would not yield the correct slope and y-intercept.

The Correct Equation: A Visual Confirmation

The correct equation, y = (-2/3)x + 4, perfectly captures the line's characteristics. The negative slope (-2/3) indicates a downward trend, and the y-intercept of 4 ensures the line crosses the y-axis at the point (0,4). If you were to graph this equation, you'd see a beautiful line gracefully descending across the coordinate plane, just as we predicted.

Real-World Applications of Linear Equations

Linear equations aren't just abstract mathematical concepts; they're powerful tools that help us understand and model the world around us. From calculating the distance traveled at a constant speed to predicting the growth of a population, linear equations are essential in various fields, including:

  • Physics: Describing motion, forces, and energy.
  • Economics: Modeling supply and demand, and analyzing financial trends.
  • Computer Science: Creating algorithms and graphics.
  • Engineering: Designing structures and systems.

The ability to identify and manipulate linear equations is a valuable skill that opens doors to countless opportunities.

Mastering Linear Equations: Tips and Tricks

To truly master linear equations, here are a few tips and tricks to keep in mind:

  • Practice, practice, practice! The more you work with linear equations, the more comfortable you'll become.
  • Visualize the graphs. Understanding how the slope and y-intercept affect the line's appearance is crucial.
  • Connect to real-world examples. Thinking about practical applications can make the concepts more tangible.
  • Don't be afraid to ask for help. If you're struggling, reach out to teachers, classmates, or online resources.

Conclusion: The Power of Slope-Intercept Form

In conclusion, the equation that perfectly describes the line with a slope of -2/3 and a y-intercept of (0,4) is y = (-2/3)x + 4. This equation, derived using the slope-intercept form, exemplifies the elegance and power of mathematical tools in describing the world around us. By understanding the slope-intercept form and practicing problem-solving, you can confidently tackle any linear equation challenge that comes your way. Keep exploring, keep learning, and keep unlocking the wonders of mathematics!

Which equation represents the line with a slope of -2/3 that intersects the y-axis at the point (0,4)?

Find the Line Equation Slope -2/3 and Y-intercept (0,4) Explained