Determine The X And Y Intercepts Of The Equation 2x + 5y = 10.

Finding the x and y intercepts of a linear equation is a fundamental concept in algebra and coordinate geometry. These intercepts represent the points where the line crosses the x-axis and the y-axis, respectively. Understanding how to calculate these intercepts is crucial for graphing linear equations and solving related problems. In this article, we will explore a step-by-step approach to finding the x and y intercepts of the equation ${ 2x + 5y = 10 }$. This process involves setting one variable to zero and solving for the other, providing valuable insights into the behavior and graphical representation of the line.

Understanding Intercepts

Before diving into the calculations, let's clarify what intercepts are and why they are important. In the Cartesian coordinate system, the x-axis is the horizontal line, and the y-axis is the vertical line. A point on the x-axis has a y-coordinate of 0, and a point on the y-axis has an x-coordinate of 0. The x-intercept is the point where the line intersects the x-axis, and the y-intercept is the point where the line intersects the y-axis. These intercepts give us two specific points on the line, which are essential for graphing the line and understanding its orientation.

Graphing a linear equation becomes much easier when you know the intercepts. By plotting the x and y intercepts on the coordinate plane, you can draw a straight line that passes through these two points. This method is a quick and efficient way to visualize the equation and its solutions. Moreover, intercepts are useful in real-world applications, such as determining the starting point and rate of change in a linear relationship. For example, in economics, the intercepts can represent the initial cost and the break-even point in a cost-benefit analysis. In physics, they might indicate the initial position and time of an object in motion.

Finding the X-Intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always 0. Therefore, to find the x-intercept of the equation ${ 2x + 5y = 10 }$, we set ${ y = 0 }$ and solve for ${ x }$. This substitution simplifies the equation, allowing us to isolate ${ x }$ and determine its value at the x-intercept. By understanding this process, you can apply it to any linear equation to find its x-intercept.

Step-by-Step Calculation

1. Set ${ y = 0 }$ in the equation: ${ 2x + 5(0) = 10 }$
2. Simplify the equation: ${ 2x + 0 = 10 }$, which becomes ${ 2x = 10 }$
3. Solve for ${ x }$ by dividing both sides by 2: ${ x = \frac{10}{2} }$
4. Calculate the value of ${ x }$: ${ x = 5 }$

Thus, the x-intercept is the point ${ (5, 0) }$. This means the line crosses the x-axis at the point where ${ x = 5 }$ and ${ y = 0 }$. This single calculation provides a critical point for graphing the line and understanding its behavior. The x-intercept is not just a mathematical point; it often has a real-world interpretation, such as the time it takes to deplete a resource or the quantity at which a cost is fully covered.

Finding the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0. To find the y-intercept of the equation ${ 2x + 5y = 10 }$, we set ${ x = 0 }$ and solve for ${ y }$. This process is analogous to finding the x-intercept but focuses on the y-coordinate. By understanding how to isolate and solve for ${ y }$, you can efficiently determine the y-intercept of any linear equation.

Step-by-Step Calculation

1. Set ${ x = 0 }$ in the equation: ${ 2(0) + 5y = 10 }$
2. Simplify the equation: ${ 0 + 5y = 10 }$, which becomes ${ 5y = 10 }$
3. Solve for ${ y }$ by dividing both sides by 5: ${ y = \frac{10}{5} }$
4. Calculate the value of ${ y }$: ${ y = 2 }$

Therefore, the y-intercept is the point ${ (0, 2) }$. This means the line crosses the y-axis at the point where ${ x = 0 }$ and ${ y = 2 }$. Like the x-intercept, the y-intercept is crucial for graphing the line and can also represent a significant value in real-world scenarios, such as the initial investment in a project or the starting amount of a substance in a chemical reaction. Knowing both intercepts gives a clear picture of the line's position and orientation on the coordinate plane.

Graphing the Line

With the x-intercept ${ (5, 0) }$ and the y-intercept ${ (0, 2) }$ determined, graphing the line becomes a straightforward process. These two points provide enough information to accurately draw the line on the Cartesian coordinate system. By connecting these points, we can visualize the linear equation and its infinite set of solutions. Graphing not only helps in understanding the equation but also in solving related problems, such as finding the intersection of two lines or determining the region defined by inequalities.

Steps to Graph

1. Plot the intercepts: Locate and mark the points ${ (5, 0) }$ and ${ (0, 2) }$ on the coordinate plane. The point ${ (5, 0) }$ is on the x-axis, 5 units to the right of the origin, and the point ${ (0, 2) }$ is on the y-axis, 2 units above the origin.
2. Draw a straight line: Use a ruler or straightedge to draw a line that passes through both plotted points. Extend the line beyond the points to cover the coordinate plane. This line represents the equation ${ 2x + 5y = 10 }$.
3. Verify the line: To ensure the line is correctly drawn, you can choose another point on the line and check if it satisfies the equation. For example, the point ${ (2.5, 1) }$ lies on the line. Substituting these values into the equation ${ 2x + 5y = 10 }$ gives ${ 2(2.5) + 5(1) = 5 + 5 = 10 }$, which confirms that the point is on the line.

The graph provides a visual representation of the equation's solutions. Every point on the line is a solution to the equation ${ 2x + 5y = 10 }$. This graphical approach is invaluable for understanding linear equations and their applications in various fields.

Real-World Applications

The concept of x and y intercepts extends beyond the classroom and has numerous real-world applications. These intercepts can represent critical values and starting points in various scenarios. Understanding how to find and interpret intercepts allows for better problem-solving and decision-making in practical situations.

Examples of Applications

1. Economics: In a supply and demand model, the intercepts can represent the price at which there is no demand (y-intercept) or the quantity supplied at no cost (x-intercept). In cost analysis, the y-intercept might represent the fixed costs, while the x-intercept could indicate the break-even point.
2. Physics: In kinematics, the intercepts can represent initial conditions. For example, in a distance-time graph, the y-intercept might indicate the initial position of an object, and the x-intercept might represent the time when the object reaches a certain point.
3. Finance: In a savings or loan scenario, the y-intercept could represent the initial investment or loan amount, and the x-intercept might represent the time it takes to pay off the loan or reach a savings goal.
4. Environmental Science: In population models, the intercepts can represent the initial population size and the carrying capacity of the environment.

By understanding the context and interpreting the intercepts, we can gain valuable insights into the relationships between variables and make informed decisions. For instance, knowing the break-even point in a business can help in setting pricing strategies and production targets. Similarly, understanding the initial conditions in a physics problem is crucial for predicting the future state of a system.

Conclusion

In conclusion, finding the x and y intercepts of the equation ${ 2x + 5y = 10 }$ is a fundamental skill in algebra with significant applications. By setting ${ y = 0 }$ to find the x-intercept and ${ x = 0 }$ to find the y-intercept, we determined that the line crosses the x-axis at ${ (5, 0) }$ and the y-axis at ${ (0, 2) }$. These intercepts not only help in graphing the line but also provide crucial information about the equation's behavior and real-world implications.

Understanding and applying the concept of intercepts is essential for solving a wide range of problems in mathematics and beyond. Whether it's graphing linear equations, analyzing economic models, or understanding physical phenomena, the ability to find and interpret intercepts is a valuable asset. Mastering this skill provides a solid foundation for more advanced mathematical concepts and practical problem-solving.