Solving 2x + Y = 6 Finding Coordinates And Sketching Graphs

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Hey guys! Let's dive into a cool math problem together. We're given the equation 2x + y = 6 and our mission is to find the solutions and sketch its graph when x takes on specific values: -3, -2, -1, 0, 1, 2, and 3. This is a classic algebra problem that combines equation solving with graphical representation. So, buckle up and let's get started!

Determining Coordinate Solutions

The first step is to find the coordinate solutions. What does that even mean? Well, a coordinate solution is simply a pair of numbers, an x-value and a y-value, that make the equation true. Since we’re given specific x-values, our job is to plug each one into the equation 2x + y = 6 and solve for the corresponding y-value. This process will give us a set of (x, y) coordinates that we can then use to sketch the graph.

Let's break it down step-by-step:

  1. When x = -3:

    Plug x = -3 into the equation: 2(-3) + y = 6

    Simplify: -6 + y = 6

    Add 6 to both sides: y = 12

    So, our first coordinate solution is (-3, 12).

  2. When x = -2:

    Plug x = -2 into the equation: 2(-2) + y = 6

    Simplify: -4 + y = 6

    Add 4 to both sides: y = 10

    Our second coordinate solution is (-2, 10).

  3. When x = -1:

    Plug x = -1 into the equation: 2(-1) + y = 6

    Simplify: -2 + y = 6

    Add 2 to both sides: y = 8

    The third solution is (-1, 8).

  4. When x = 0:

    Substitute x = 0: 2(0) + y = 6

    Simplify: 0 + y = 6

    Therefore, y = 6

    The coordinate solution here is (0, 6).

  5. When x = 1:

    Substitute x = 1: 2(1) + y = 6

    Simplify: 2 + y = 6

    Subtract 2 from both sides: y = 4

    So, we have the solution (1, 4).

  6. When x = 2:

    Substitute x = 2: 2(2) + y = 6

    Simplify: 4 + y = 6

    Subtract 4 from both sides: y = 2

    The solution is (2, 2).

  7. When x = 3:

    Substitute x = 3: 2(3) + y = 6

    Simplify: 6 + y = 6

    Subtract 6 from both sides: y = 0

    Finally, we have the solution (3, 0).

So, to recap, we have found the following coordinate solutions:

  • (-3, 12)
  • (-2, 10)
  • (-1, 8)
  • (0, 6)
  • (1, 4)
  • (2, 2)
  • (3, 0)

These are the points we'll use to sketch the graph. Each pair of numbers represents a point on the coordinate plane, where the first number is the x-coordinate and the second number is the y-coordinate. We've systematically gone through each given x-value, plugged it into our equation, and solved for the corresponding y-value. This is a fundamental technique in algebra for finding solutions to linear equations.

Sketching the Graph

Now comes the fun part – sketching the graph! We'll use the coordinate solutions we just found to draw a visual representation of the equation 2x + y = 6. A graph gives us a clear picture of how x and y are related.

To sketch the graph, we’ll follow these steps:

  1. Draw the Coordinate Plane: First, we need to draw our x and y axes. This is basically two number lines that intersect at a 90-degree angle. The horizontal line is the x-axis, and the vertical line is the y-axis. The point where they meet is called the origin, and it represents the point (0, 0).

  2. Plot the Points: Now, we'll plot each of the coordinate solutions we calculated earlier. Remember, each point is written in the form (x, y), where x tells us how far to move horizontally from the origin (left if x is negative, right if x is positive), and y tells us how far to move vertically (down if y is negative, up if y is positive).

    • Let's plot (-3, 12). Start at the origin, move 3 units to the left along the x-axis, and then 12 units up along the y-axis. Mark this point.
    • Next, plot (-2, 10). Move 2 units left and 10 units up. Mark this point.
    • Continue plotting the remaining points: (-1, 8), (0, 6), (1, 4), (2, 2), and (3, 0).

    It’s super important to be accurate when plotting points. A slight mistake in plotting can lead to an incorrect graph, and we want our visual representation to be spot on!

  3. Draw the Line: Once all the points are plotted, you'll notice that they form a straight line. This is because the equation 2x + y = 6 is a linear equation. This means that the relationship between x and y is constant, and when we graph it, we get a straight line. Take a ruler or a straightedge and draw a line that passes through all the points you've plotted. Make sure the line extends beyond the points on both ends, indicating that the line continues infinitely in both directions.

And there you have it! You've successfully sketched the graph of the equation 2x + y = 6 for the given values of x. The line you've drawn represents all the possible solutions to the equation. Any point on this line will satisfy the equation 2x + y = 6. This is the beauty of graphical representation – it gives us a visual way to understand the solutions of an equation.

Understanding the Significance

So, why is this important? Understanding how to find solutions and sketch graphs is a fundamental skill in algebra and mathematics in general. Linear equations and their graphs pop up everywhere in real-world applications, from calculating distances and speeds to modeling financial trends. By mastering these basics, you're laying a solid foundation for more advanced math concepts and problem-solving.

Let's think about it a little more. What does the graph actually tell us? Each point on the line represents a solution to the equation. If you pick any point on the line and plug its x and y coordinates into the equation 2x + y = 6, the equation will hold true. The line is a visual representation of all the possible solutions.

Also, notice the slope of the line. The line slopes downwards from left to right. This indicates that as x increases, y decreases. The slope tells us how steep the line is and in what direction it's going. We can even calculate the slope using any two points on the line. The slope, often denoted as 'm', can be found using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

In our case, let's take the points (0, 6) and (1, 4). Plugging these into the formula, we get:

m = (4 - 6) / (1 - 0) = -2 / 1 = -2

So, the slope of our line is -2. This means that for every 1 unit increase in x, y decreases by 2 units. The slope gives us valuable information about the relationship between x and y in the equation.

Conclusion

Alright, guys, we've done a fantastic job today! We successfully found the coordinate solutions for the equation 2x + y = 6 when x takes on the values -3, -2, -1, 0, 1, 2, and 3. Then, we used those solutions to sketch the graph of the equation. We also discussed the significance of the graph and how it represents all the possible solutions to the equation. We even touched on the concept of slope and how it helps us understand the relationship between x and y.

Remember, practice makes perfect. The more you work with equations and graphs, the more comfortable you'll become with these concepts. So, keep practicing, keep exploring, and keep having fun with math! You've got this!

To further solidify your understanding, let's tackle some frequently asked questions related to solving equations and sketching graphs:

1. What is a coordinate solution?

A coordinate solution is a pair of values (x, y) that, when substituted into an equation, make the equation true. It represents a point on the graph of the equation. Finding coordinate solutions is a crucial step in sketching the graph of an equation because these solutions give us the exact points that lie on the line or curve.

2. How do I find coordinate solutions for a linear equation?

To find coordinate solutions for a linear equation, you typically substitute specific values for one variable (usually x) into the equation and solve for the other variable (y). Each pair of values (x, y) that you find in this way is a coordinate solution. For instance, in our example, we substituted the given values of x (-3, -2, -1, 0, 1, 2, 3) into the equation 2x + y = 6 and solved for the corresponding y-values.

3. What does the graph of a linear equation represent?

The graph of a linear equation is a straight line that represents all the possible solutions to the equation. Each point on the line corresponds to a coordinate solution (x, y) that satisfies the equation. When you sketch the graph, you are essentially creating a visual representation of the relationship between the variables x and y.

4. Why is it important to plot points accurately when sketching a graph?

Accurate plotting of points is essential because it directly affects the accuracy of the graph. If points are plotted incorrectly, the resulting line may not accurately represent the equation, leading to misinterpretations and incorrect solutions. A precise graph is a reliable tool for understanding and analyzing the equation's behavior.

5. What is the significance of the slope of a line?

The slope of a line is a measure of its steepness and direction. It tells us how much the y-value changes for every unit change in the x-value. A positive slope indicates that the line slopes upwards from left to right, while a negative slope indicates that the line slopes downwards. The slope provides valuable information about the rate of change and the relationship between the variables in the equation.

6. How do you calculate the slope of a line?

The slope (m) of a line can be calculated using any two points on the line, (x1, y1) and (x2, y2), using the formula: m = (y2 - y1) / (x2 - x1). This formula essentially calculates the change in y divided by the change in x, giving us the rate of change. In our example, we calculated the slope using the points (0, 6) and (1, 4) and found it to be -2.

7. What is the difference between a linear and a non-linear equation?

A linear equation is an equation that, when graphed, forms a straight line. It typically involves variables raised to the power of 1. A non-linear equation, on the other hand, does not form a straight line when graphed. Non-linear equations can involve variables raised to powers other than 1, or other mathematical functions like exponential or trigonometric functions. The methods for solving and graphing non-linear equations can be more complex than those for linear equations.

8. Can I use any two points on the line to calculate the slope?

Yes, you can use any two distinct points on the line to calculate the slope. The slope of a straight line is constant, meaning it is the same between any two points on the line. Therefore, regardless of which two points you choose, the calculated slope should be the same. This consistency is a fundamental property of linear equations and their graphs.

9. What are some real-world applications of linear equations and graphs?

Linear equations and graphs have numerous real-world applications across various fields. They are used in physics to describe motion and forces, in economics to model supply and demand, in finance to calculate interest and investments, and in many other areas. Understanding linear equations is crucial for problem-solving and decision-making in many practical situations. For example, they can be used to predict future values, optimize resources, and analyze trends.

10. How does the y-intercept relate to the graph of a linear equation?

The y-intercept is the point where the graph of the equation intersects the y-axis. It is the value of y when x is equal to 0. In the equation 2x + y = 6, we found that when x = 0, y = 6. So, the y-intercept is the point (0, 6). The y-intercept is a key feature of the graph because it gives us a starting point on the y-axis and helps us visualize the line's position on the coordinate plane. It is often denoted by the variable 'b' in the slope-intercept form of a linear equation, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Hopefully, these FAQs have shed more light on the topic and clarified any remaining questions you might have. Remember, math is a journey, and with each step, you're building a stronger understanding of the world around you. Keep exploring and keep learning!

For quick reference, here’s a summary of the coordinate solutions we found for the equation 2x + y = 6 when x = -3, -2, -1, 0, 1, 2, and 3:

  • x = -3, y = 12; Coordinate Solution: (-3, 12)
  • x = -2, y = 10; Coordinate Solution: (-2, 10)
  • x = -1, y = 8; Coordinate Solution: (-1, 8)
  • x = 0, y = 6; Coordinate Solution: (0, 6)
  • x = 1, y = 4; Coordinate Solution: (1, 4)
  • x = 2, y = 2; Coordinate Solution: (2, 2)
  • x = 3, y = 0; Coordinate Solution: (3, 0)

These coordinate pairs are the backbone of our graph. Each pair represents a point that lies on the line 2x + y = 6. When we plotted these points and connected them, we created a visual representation of the equation, making it easier to understand the relationship between x and y.

Keep this summary handy as you continue to explore linear equations and graphs. It's a great reminder of the practical steps involved in finding solutions and bringing them to life visually. Remember, every equation tells a story, and graphs are the illustrations that help us see that story clearly.

Happy graphing, everyone! And keep up the amazing work!