Solving Systems Of Equations -2x+y=10 And Y=(x-1)^2+4

Introduction: Understanding Systems of Equations

When dealing with mathematical problems, we often encounter systems of equations. In this comprehensive guide, we will dive deep into the solutions for a specific system, which includes a linear equation and a quadratic equation. Understanding the solutions to these systems is crucial for various applications in mathematics, physics, engineering, and other fields. Our focus will be on the system defined by the equations -2x + y = 10 and y = (x - 1)^2 + 4. We will explore different methods to find the solutions, including substitution and graphical approaches. This journey will not only provide the answers but also enhance your problem-solving skills in algebra and coordinate geometry.

The Equations: A Detailed Look

At the heart of our exploration are two equations that define our system. The first equation, -2x + y = 10, is a linear equation. Linear equations represent straight lines when graphed on a coordinate plane. Understanding this linearity is key to visualizing and solving the system. We can rewrite this equation in slope-intercept form, y = 2x + 10, to easily identify its slope and y-intercept. The slope, which is 2, tells us how steeply the line rises or falls, and the y-intercept, which is 10, indicates where the line crosses the vertical axis. This form makes it straightforward to graph the line and analyze its behavior.

The second equation, y = (x - 1)^2 + 4, introduces a different type of curve: a parabola. This is a quadratic equation, and its graph is U-shaped. The vertex form of a parabola, y = a(x - h)^2 + k, provides significant insights. In our case, the vertex is at the point (1, 4), which is the lowest point on the curve since the coefficient of the squared term is positive. The parabola opens upwards, and its symmetry around the vertical line x = 1 is an important characteristic. Recognizing the shape and key features of this parabola is essential for solving the system graphically and algebraically.

Solving by Substitution: An Algebraic Approach

One of the most effective methods for solving systems of equations is substitution. This technique involves solving one equation for one variable and then substituting that expression into the other equation. For our system, the linear equation -2x + y = 10 can be easily rearranged to isolate y: y = 2x + 10. Now, we substitute this expression for y into the quadratic equation y = (x - 1)^2 + 4. This gives us 2x + 10 = (x - 1)^2 + 4.

Expanding and simplifying the equation, we get 2x + 10 = x^2 - 2x + 1 + 4. Combining like terms and setting the equation to zero, we obtain a quadratic equation in standard form: x^2 - 4x - 5 = 0. This quadratic equation can be solved using factoring, completing the square, or the quadratic formula. Factoring is often the quickest method if the equation factors easily. In this case, we can factor the quadratic as (x - 5)(x + 1) = 0. This gives us two possible solutions for x: x = 5 and x = -1.

To find the corresponding y values, we substitute each x value back into either of the original equations. Using the linear equation y = 2x + 10, when x = 5, y = 2(5) + 10 = 20. When x = -1, y = 2(-1) + 10 = 8. Therefore, the solutions to the system of equations are the points (5, 20) and (-1, 8). These points represent the intersections of the line and the parabola on the coordinate plane.

Graphical Solutions: Visualizing the Intersections

A graphical approach provides a visual understanding of the solutions to our system of equations. By plotting both the line and the parabola on the same coordinate plane, we can identify the points where they intersect. These intersection points represent the solutions to the system, as they satisfy both equations simultaneously.

The linear equation -2x + y = 10 or y = 2x + 10 can be graphed by plotting its y-intercept (10) and using its slope (2) to find additional points. For example, moving one unit to the right from the y-intercept, we move two units up to find another point on the line. Connecting these points gives us the graph of the line.

The quadratic equation y = (x - 1)^2 + 4 represents a parabola with its vertex at (1, 4). We can plot additional points by substituting different x values into the equation and calculating the corresponding y values. For example, when x = 0, y = (0 - 1)^2 + 4 = 5, giving us the point (0, 5). Similarly, when x = 2, y = (2 - 1)^2 + 4 = 5, giving us the point (2, 5). By plotting these points and connecting them smoothly, we can draw the parabola.

When we graph both the line and the parabola, we will observe that they intersect at two points. These points are (5, 20) and (-1, 8), which confirms our algebraic solution. The graphical method not only provides the solutions but also enhances our understanding of how the line and the parabola interact.

Real-World Applications: Why These Solutions Matter

The solutions to systems of equations have real-world applications across various fields. In physics, these systems can model the intersection of trajectories, such as the path of a projectile and a target. In economics, they can determine equilibrium points in supply and demand models. In engineering, they can be used to analyze structural designs and optimize performance. The ability to solve these systems is a valuable skill in any quantitative field.

For example, consider a scenario where a company's cost function is represented by a quadratic equation, and its revenue function is represented by a linear equation. The points of intersection, which are the solutions to the system, represent the break-even points where the company's costs equal its revenue. Understanding these points is crucial for making informed business decisions.

In physics, the intersection points could represent the time and position at which two objects collide. By setting up and solving a system of equations that describe the motion of each object, we can predict when and where they will meet. This has applications in areas such as air traffic control and satellite tracking.

Conclusion: Mastering Systems of Equations

In this guide, we have thoroughly explored the solutions to the system of equations -2x + y = 10 and y = (x - 1)^2 + 4. We used both algebraic (substitution) and graphical methods to find the solutions, which are the points (5, 20) and (-1, 8). Understanding these methods is crucial for solving a wide range of mathematical problems and real-world applications.

Mastering systems of equations is a fundamental skill in mathematics. It requires a solid understanding of linear and quadratic equations, as well as the ability to manipulate and solve them. By practicing these techniques and applying them to different problems, you can enhance your problem-solving skills and deepen your understanding of mathematical concepts. Whether you are a student, engineer, or anyone dealing with quantitative analysis, the ability to solve systems of equations will prove to be an invaluable asset.