Solving Simultaneous Equations X - Y = 6 And Y = 2x - 8

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In the realm of mathematics, solving simultaneous equations is a fundamental skill. These equations, also known as systems of equations, involve two or more equations with a shared set of variables. The solution to a system of equations represents the values of the variables that satisfy all equations simultaneously. In this comprehensive guide, we will delve into the process of solving the pair of simultaneous equations: x - y = 6 and y = 2x - 8. We will explore different methods, discuss the underlying concepts, and provide a step-by-step solution to help you grasp this essential mathematical concept.

Understanding Simultaneous Equations

Before we dive into the solution, let's establish a solid understanding of simultaneous equations. A system of equations is a set of two or more equations that share the same variables. The goal is to find the values of these variables that make all the equations true at the same time. This intersection point, where all equations hold true, represents the solution to the system. Simultaneous equations are a cornerstone of various mathematical and scientific disciplines, including algebra, calculus, physics, and economics. They are used to model real-world scenarios involving multiple interconnected variables, allowing us to make predictions and solve complex problems. Understanding simultaneous equations is crucial for anyone pursuing further studies in STEM fields or seeking to apply mathematical principles to practical situations.

Methods for Solving Simultaneous Equations

There are several methods available to solve simultaneous equations, each with its own advantages and disadvantages. We will focus on two primary methods: substitution and elimination. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved directly. The elimination method, on the other hand, involves manipulating the equations to eliminate one of the variables. This is typically achieved by multiplying one or both equations by constants so that the coefficients of one variable are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one variable.

Substitution Method

The substitution method is a powerful technique for solving simultaneous equations, particularly when one equation is already solved for one variable or can be easily manipulated to do so. The core idea behind this method is to express one variable in terms of the other, allowing us to reduce the system of equations to a single equation with a single variable. This simplified equation can then be solved directly, and the solution can be substituted back into one of the original equations to find the value of the other variable. The substitution method is particularly useful when dealing with non-linear equations or systems with more than two variables. It provides a systematic approach to solving complex systems, making it a valuable tool in various mathematical and scientific applications.

Elimination Method

The elimination method is another widely used technique for solving simultaneous equations. This method focuses on eliminating one of the variables by manipulating the equations so that the coefficients of one variable are opposites. This is typically achieved by multiplying one or both equations by suitable constants. Once the coefficients are opposites, adding the equations together will eliminate that variable, leaving a single equation with one variable. This equation can then be solved, and the solution can be substituted back into one of the original equations to find the value of the other variable. The elimination method is particularly efficient when the coefficients of one variable are easily made opposites, making it a preferred choice for certain types of systems of equations.

Solving the Equations x - y = 6 and y = 2x - 8

Now, let's apply these methods to solve the given pair of simultaneous equations: x - y = 6 and y = 2x - 8. We will demonstrate both the substitution and elimination methods to provide a comprehensive understanding of the solution process.

Using the Substitution Method

  1. Identify an equation to solve for one variable: In this case, the second equation, y = 2x - 8, is already solved for y.
  2. Substitute the expression for y into the other equation: Substitute 2x - 8 for y in the first equation, x - y = 6. This gives us: x - (2x - 8) = 6
  3. Simplify and solve for x: Distribute the negative sign: x - 2x + 8 = 6. Combine like terms: -x + 8 = 6. Subtract 8 from both sides: -x = -2. Divide both sides by -1: x = 2
  4. Substitute the value of x back into either equation to solve for y: Substitute x = 2 into the equation y = 2x - 8: y = 2(2) - 8. Simplify: y = 4 - 8. Therefore, y = -4

Thus, using the substitution method, we find that the solution to the system of equations is x = 2 and y = -4.

Using the Elimination Method

  1. Align the equations: Write the equations one below the other, aligning the x and y terms:
    • x - y = 6
    • y = 2x - 8. Rewrite the second equation to align the terms: -2x + y = -8
  2. Multiply one or both equations by a constant so that the coefficients of one variable are opposites: Notice that the y terms already have opposite signs. The coefficients are -1 and 1.
  3. Add the equations together: Add the two equations:
    • x - y = 6
    • -2x + y = -8 This results in: (x - 2x) + (-y + y) = 6 - 8, which simplifies to -x = -2
  4. Solve for x: Divide both sides by -1: x = 2
  5. Substitute the value of x back into either equation to solve for y: Substitute x = 2 into the equation x - y = 6: 2 - y = 6. Subtract 2 from both sides: -y = 4. Divide both sides by -1: y = -4

Using the elimination method, we also arrive at the solution x = 2 and y = -4.

Verifying the Solution

It's always a good practice to verify the solution by substituting the values of x and y back into the original equations. This ensures that the solution satisfies both equations simultaneously.

For the first equation, x - y = 6, substitute x = 2 and y = -4: 2 - (-4) = 6, which simplifies to 2 + 4 = 6, which is true.

For the second equation, y = 2x - 8, substitute x = 2 and y = -4: -4 = 2(2) - 8, which simplifies to -4 = 4 - 8, which is also true.

Since the solution x = 2 and y = -4 satisfies both equations, we can confidently conclude that it is the correct solution to the system of equations.

Conclusion

In this comprehensive guide, we have explored the process of solving the pair of simultaneous equations x - y = 6 and y = 2x - 8. We have discussed the fundamental concepts of simultaneous equations, the substitution method, and the elimination method. By applying both methods, we have arrived at the solution x = 2 and y = -4. We have also emphasized the importance of verifying the solution to ensure its accuracy. Solving simultaneous equations is a crucial skill in mathematics and its applications, and we hope this guide has provided you with a clear and thorough understanding of the process.