Finding The Equation Of A Line With Y-Intercept 2/3 And Slope -3

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Hey guys! Today, we're diving into the world of linear equations and tackling a fun problem. We're going to figure out which equation represents a line that crosses the y-axis at 2/3 and has a slope of -3. It might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super clear. So, grab your thinking caps, and let's get started!

Understanding Slope-Intercept Form

When we're dealing with linear equations, one of the handiest forms to know is the slope-intercept form. This form is written as:

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, which 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.
  • b is the y-intercept, which is the point where the line crosses the y-axis. In other words, it's the value of y when x is 0.

Think of the slope as the rise over run. It tells you how much the y-value changes for every one unit change in the x-value. A slope of -3, for example, means that for every one unit you move to the right on the graph, the line goes down by 3 units. The y-intercept, on the other hand, is simply where the line starts on the vertical axis. It's like the line's home base on the y-axis.

Knowing this form is super useful because it lets us quickly identify the slope and y-intercept of a line just by looking at its equation. It's like having a secret decoder ring for linear equations!

Applying Slope-Intercept Form to Our Problem

In our problem, we're given a y-intercept of 2/3 and a slope of -3. So, we can directly plug these values into the slope-intercept form:

y = mx + b

y = (-3)x + (2/3)

y = -3x + 2/3

This equation, y = -3x + 2/3, represents a line with the slope and y-intercept we're looking for. However, the answer choices are given in a slightly different form, called the standard form. So, we need to do a little bit of algebraic maneuvering to get our equation into the standard form and see which answer choice matches.

Converting to Standard Form

The standard form of a linear equation is written as:

Ax + By = C

Where A, B, and C are constants, and A is usually a positive integer. To convert our equation from slope-intercept form to standard form, we need to get rid of the fraction and rearrange the terms.

Here's how we can do it:

  1. Get rid of the fraction: To eliminate the fraction (2/3), we can multiply both sides of the equation by 3:

    • 3*(y) = 3*(-3x + 2/3)
    • 3y = -9x + 2

  2. Rearrange the terms: Now, we want to get the x and y terms on the same side of the equation. We can do this by adding 9x to both sides:

    • 9x + 3y = -9x + 9x + 2
    • 9x + 3y = 2

And there you have it! Our equation is now in standard form: 9x + 3y = 2. This matches one of the answer choices, which means we've found the correct equation.

Analyzing the Answer Choices

Now, let's take a look at the answer choices and see how our solution lines up:

  • A. 2x - 3y = 9
  • B. 2x + 3y = 9
  • C. 9x - 3y = 2
  • D. 9x + 3y = 2

We found that the equation representing a line with a y-intercept of 2/3 and a slope of -3 is 9x + 3y = 2. Comparing this to the answer choices, we can see that option D is the correct answer.

To further solidify our understanding, let's analyze why the other options are incorrect:

  • Option A (2x - 3y = 9): To find the slope and y-intercept of this equation, we'd need to convert it to slope-intercept form. If we did that, we'd find that the slope and y-intercept don't match our requirements.
  • Option B (2x + 3y = 9): Similar to option A, converting this equation to slope-intercept form would reveal that it doesn't have the correct slope and y-intercept.
  • Option C (9x - 3y = 2): This equation has the correct constant term (2), but the sign in front of the y term is different. This would result in a different slope when converted to slope-intercept form.

By carefully analyzing each answer choice, we can confidently confirm that option D is the only equation that satisfies the given conditions.

Graphing the Line (Optional)

For those who are visually inclined, it can be helpful to graph the line to see if it matches our expectations. We can graph the line 9x + 3y = 2 by finding two points on the line and drawing a line through them.

One easy point to find is the y-intercept. We already know that the y-intercept is 2/3, which is approximately 0.67. So, one point on the line is (0, 0.67).

To find another point, we can choose any value for x and solve for y. For example, let's choose x = 1:

  • 9(1) + 3y = 2
  • 9 + 3y = 2
  • 3y = -7
  • y = -7/3*, which is approximately -2.33

So, another point on the line is (1, -2.33).

If we plot these two points and draw a line through them, we'll see that the line has a negative slope and crosses the y-axis at 2/3, confirming our solution.

Key Takeaways and Real-World Applications

Alright, guys, let's recap what we've learned and see why this stuff actually matters in the real world.

Key Takeaways

  • Slope-intercept form (y = mx + b): This is your best friend when you need to quickly identify the slope and y-intercept of a line. m is the slope, and b is the y-intercept.
  • Standard form (Ax + By = C): Another way to write linear equations. We can convert between slope-intercept form and standard form using algebraic manipulation.
  • Solving for unknowns: We used our knowledge of linear equations to find the equation that matches specific criteria (y-intercept and slope). This is a fundamental skill in algebra and beyond.
  • Analyzing answer choices: Don't just pick the first answer that looks right! Take the time to analyze why the other options are incorrect. This helps solidify your understanding and prevents careless errors.

Real-World Applications

You might be wondering, "Okay, this is cool, but when am I ever going to use this in real life?" Well, linear equations are everywhere! Here are just a few examples:

  • Finance: Calculating loan payments, investment growth, and budgeting often involves linear relationships. The slope might represent the interest rate, and the y-intercept could be the initial investment.
  • Physics: Describing motion at a constant speed. The slope could be the velocity, and the y-intercept might be the initial position.
  • Engineering: Designing structures, circuits, and systems often requires understanding linear relationships between variables. For example, the relationship between voltage and current in a resistor is linear.
  • Data analysis: Linear regression is a powerful tool for finding trends in data. We can use linear equations to model the relationship between two variables and make predictions.
  • Everyday life: Calculating the cost of a taxi ride (initial fare + cost per mile), determining the amount of paint needed for a room (area to be painted), or even figuring out how much time it will take to drive somewhere (distance/speed). All of these scenarios can involve linear equations.

So, the next time you're faced with a problem involving a constant rate of change or a starting value, remember the power of linear equations! They're a fundamental tool for understanding and solving problems in a wide range of fields.

Practice Problems

To really master this stuff, practice is key! Here are a few practice problems you can try:

  1. Find the equation of a line with a slope of 2 and a y-intercept of -1.
  2. What is the slope and y-intercept of the line 3x - 4y = 12?
  3. Write the equation y = (2/5)x + 3 in standard form.

Try solving these problems on your own, and then check your answers with a friend or teacher. The more you practice, the more confident you'll become in your ability to work with linear equations.

Conclusion

Great job, everyone! We successfully decoded the problem of finding the equation with a specific y-intercept and slope. We learned how to use slope-intercept form, convert to standard form, and analyze answer choices. Remember, linear equations are a fundamental concept in math and have tons of real-world applications. So, keep practicing, and you'll be a pro in no time!

If you have any questions or want to explore more about linear equations, feel free to ask. Keep up the awesome work, guys, and I'll catch you in the next math adventure!