Solving For X And Y A Mathematical Exploration

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In mathematics, we often encounter problems that require us to solve for unknown variables. These problems can range from simple equations to complex systems of equations. Mastering these skills is crucial for success in various fields, including engineering, physics, economics, and computer science. This article delves into a specific problem involving two variables, x and y, and a system of two equations. We will explore the steps involved in solving this system and ultimately finding the product of x and y. This exploration will not only enhance your problem-solving skills but also provide a deeper understanding of mathematical concepts.

Problem Statement Unveiling the Equations

Our journey begins with a problem statement that presents us with two key pieces of information. First, we are told that the sum of a number x and twice a number y is -7. This can be translated into a mathematical equation as follows:

x + 2y = -7

This equation represents a linear relationship between x and y. It tells us that as x changes, y must also change in a way that maintains the sum of x and twice y at -7. Understanding this relationship is crucial for visualizing the possible solutions.

Second, we are given another piece of information: the difference between three times the number x and the number y is equal to 7. This can be written as:

3x - y = 7

This equation introduces another linear relationship between x and y. The difference between three times x and y must always equal 7. This provides us with a second constraint on the values of x and y.

Together, these two equations form a system of linear equations. A system of equations is a set of two or more equations that involve the same variables. The goal of solving a system of equations is to find the values of the variables that satisfy all the equations simultaneously. In our case, we need to find the values of x and y that satisfy both x + 2y = -7 and 3x - y = 7.

To solve this system, we will employ a method known as substitution, which is one of the fundamental techniques for solving systems of equations. This method involves solving one equation for one variable and then substituting that expression into the other equation. This will result in a single equation with one variable, which can be solved directly. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

Solving the System A Step-by-Step Approach

To effectively solve the system of equations, we can use several methods, including substitution, elimination, and matrix methods. In this case, the substitution method is particularly suitable. This method involves the following steps:

  1. Isolate one variable in one of the equations. Let's choose the second equation, 3x - y = 7, and solve for y. Adding y to both sides and subtracting 7 from both sides, we get:

    y = 3x - 7

    This equation expresses y in terms of x. Now we can substitute this expression for y in the first equation.

  2. Substitute the expression into the other equation. Substitute y = 3x - 7 into the first equation, x + 2y = -7:

    x + 2(3x - 7) = -7

    This step replaces y in the first equation with its equivalent expression in terms of x, creating a new equation with only one variable, x.

  3. Solve the resulting equation for the remaining variable. Now we have an equation with only one variable, x. Let's simplify and solve for x:

    x + 6x - 14 = -7

    7x - 14 = -7

    7x = 7

    x = 1

    This gives us the value of x. Now that we know x, we can find y.

  4. Substitute the value back into one of the original equations to solve for the other variable. Substitute x = 1 into the equation y = 3x - 7:

    y = 3(1) - 7

    y = 3 - 7

    y = -4

    So, we have found the values of both variables: x = 1 and y = -4.

  5. Check the solution. It is always a good practice to check our solution by substituting the values of x and y back into the original equations to ensure they hold true:

    • Equation 1: x + 2y = -7

      1 + 2(-4) = 1 - 8 = -7 (True)

    • Equation 2: 3x - y = 7

      3(1) - (-4) = 3 + 4 = 7 (True)

    Since both equations are satisfied, our solution is correct.

Product of x and y The Final Calculation

Now that we have found the values of x and y, the final step is to find their product. The problem asks us to calculate x * y.

Given x = 1 and y = -4, the product is:

x * y = 1 * (-4) = -4

Thus, the product of x and y is -4. This completes the solution to the problem. Understanding each step and the underlying mathematical principles is crucial for applying these techniques to more complex problems.

Conclusion Key Takeaways

In this article, we tackled a mathematical problem involving a system of two linear equations. We successfully solved for the values of x and y using the substitution method and then calculated their product. The key takeaways from this exploration are:

  • Understanding the problem statement: Clearly translating the given information into mathematical equations is the first and most crucial step.
  • Choosing an appropriate method: The substitution method proved to be an efficient way to solve this particular system of equations. However, other methods like elimination or matrix methods can also be used depending on the nature of the equations.
  • Step-by-step execution: Breaking down the problem into smaller, manageable steps helps in avoiding errors and ensures a clear and logical solution process.
  • Verification: Always check your solution by substituting the values back into the original equations to ensure accuracy.
  • Application of fundamental concepts: This problem reinforces the understanding of linear equations, systems of equations, and algebraic manipulation.

This exercise demonstrates the importance of problem-solving skills in mathematics. The ability to translate word problems into mathematical expressions, choose appropriate methods, and execute the steps accurately is essential for success in mathematics and related fields. By practicing these skills, we can develop a deeper understanding of mathematical concepts and enhance our ability to solve complex problems.

This exploration into solving for x and y showcases the beauty and power of mathematical thinking. By mastering these fundamental techniques, we equip ourselves to tackle a wide range of challenges in mathematics and beyond. Remember, practice is key to developing proficiency in problem-solving. So, keep exploring, keep questioning, and keep solving!