In the world of mathematics, the beauty of geometry often lies in its simplicity. Today, we embark on a visual journey, where we'll explore how to draw the graphs of two equations, X + Y = 10 and 3X + 2Y = 12, determine the coordinates of their vertices, and shade the triangular region they create with the X-axis. This article will break down the steps in a conversational style, making it easy for you to grasp the concept and apply it in your own mathematical adventures.

Understanding the Equations

Let's begin by understanding the equations we're working with:

Equation 1: X + Y = 10

This is a linear equation in two variables, X and Y. It represents a straight line on the Cartesian plane. But what does this line look like? To find out, we need to draw it.

Equation 2: 3X + 2Y = 12

Similarly, this equation also represents a straight line but with different slopes and intercepts. We'll draw this one too.

Graphing the Equations

Now that we know what we're dealing with, let's roll up our sleeves and start graphing these equations.

Drawing the Graph of X + Y = 10

To graph this equation, we'll rearrange it to find its intercepts. When X = 0, Y = 10, and when Y = 0, X = 10. We now have two points: (0, 10) and (10, 0). Plot these on your Cartesian plane and connect the dots. Voilà! You have a straight line.

Drawing the Graph of 3X + 2Y = 12

This equation can also be rearranged for intercepts. When X = 0, Y = 6, and when Y = 0, X = 4. Plot these points, connect the dots, and you've got another straight line.

Finding the Vertices

Now that we have our two lines on the graph, we can determine the vertices of the triangle formed by these lines and the X-axis.

Vertex A: (0, 0)

One vertex of the triangle is where both lines intersect the X-axis. For the first line, X + Y = 10, when Y = 0, X = 10. For the second line, 3X + 2Y = 12, when Y = 0, X = 4. So, we have Vertex A at (0, 0).

Vertex B: (4, 0)

The second vertex is where the second line intersects the X-axis, which is at (4, 0).

Vertex C: (10, 0)

The third vertex is where the first line intersects the X-axis, which is at (10, 0).

Shading the Triangular Region

Now that we've determined the vertices (A, B, and C) of our triangle, we can shade the region inside it. This represents the solution to the system of equations.

Conclusion

Drawing the graphs of equations and finding the vertices of the resulting shapes can be a fascinating exercise in geometry. In this article, we've explored how to graph two linear equations, determine their vertices, and shade the triangular region they create. Remember, geometry is not just about numbers; it's about visualizing and understanding the world around us in a mathematical way.

FAQs (Frequently Asked Questions)

1. What is the significance of finding the vertices of a shape on the X-axis?

   • Understanding vertices helps us visualize solutions to equations and inequalities, making math more tangible.

2. Can these concepts be applied to equations with more variables?

   • Yes, the principles remain the same, but the complexity increases with additional variables.

3. Why is graphing important in mathematics?

   • Graphs provide a visual representation of mathematical concepts, aiding in understanding and problem-solving.

4. Are there software tools that can graph equations for us?

   • Yes, various graphing calculators and software make this task much easier.

5. Where can I learn more about advanced geometric concepts?

   • You can explore online resources, textbooks, and math forums to delve deeper into geometry and related fields.