Solving Systems Of Equations A Step-by-Step Guide

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Let's delve into the intricacies of solving systems of equations. In this comprehensive guide, we will dissect a specific system of equations and meticulously explain the steps involved in arriving at the solution. Our focus will be on clarity and understanding, ensuring that you grasp the underlying principles and can confidently tackle similar problems. We'll start with the given system, denoted as System A:

System A

${
\begin{aligned}
x - y &= 3 \\
-2x + 4y &= -2
\end{aligned}
}$

Along with the system, we are provided with the solution: (5, 2). This means that the values x = 5 and y = 2 simultaneously satisfy both equations in the system. Our goal is to understand how this solution was obtained. We will break down the process into manageable steps, providing explanations and justifications along the way. The journey to solving a system of equations often involves strategic manipulation and a keen eye for patterns. We'll explore techniques such as elimination and substitution, which are powerful tools in our arsenal. By understanding these methods, you'll be well-equipped to solve a wide range of systems of equations. Furthermore, we will emphasize the importance of verifying the solution. Plugging the values back into the original equations is a crucial step in ensuring the accuracy of our work. It serves as a final check, giving us confidence in our answer. So, let's embark on this mathematical journey together, unraveling the steps that lead to the solution (5, 2) for System A.

The core of solving systems of equations lies in manipulating the equations in a way that eliminates one variable, allowing us to solve for the other. There are primarily two methods to achieve this: substitution and elimination. In this case, we will use the elimination method, which involves strategically multiplying equations by constants to make the coefficients of one variable opposites, thereby canceling them out when the equations are added.

Step 1: Manipulating the Equations

Our primary goal here is to modify one or both equations so that the coefficients of either x or y are additive inverses (opposites). Looking at System A, we can observe that the coefficient of x in the first equation is 1, and in the second equation, it is -2. This provides us with a clear path forward. If we multiply the entire first equation by 2, the coefficient of x will become 2, which is the opposite of -2. Let's perform this operation:

2 * (x - y) = 2 * 3

This simplifies to:

2x - 2y = 6

Now, we have a modified first equation. We'll refer to this as Equation 1':

Equation 1':

2x - 2y = 6

Our original second equation remains unchanged:

Equation 2:

-2x + 4y = -2

Step 2: Eliminating a Variable

With the equations strategically modified, we can now eliminate the x variable. To do this, we will add Equation 1' and Equation 2 together. Notice that the x terms will cancel out, leaving us with an equation in terms of y only:

(2x - 2y) + (-2x + 4y) = 6 + (-2)

Combining like terms, we get:

2y = 4

This is a significant step, as we've successfully reduced the system to a single equation with one unknown.

Step 3: Solving for y

Solving for y is now a straightforward process. We have the equation:

2y = 4

To isolate y, we simply divide both sides of the equation by 2:

y = 4 / 2

This gives us the value of y:

y = 2

So, we've determined that the y-coordinate of the solution is 2.

Step 4: Solving for x

Now that we know the value of y, we can substitute it back into either of the original equations (or Equation 1') to solve for x. Let's use the first original equation, as it appears simpler:

Equation 1:

x - y = 3

Substitute y = 2 into the equation:

x - 2 = 3

To isolate x, we add 2 to both sides of the equation:

x = 3 + 2

This gives us the value of x:

x = 5

Therefore, we've found that the x-coordinate of the solution is 5.

Step 5: Verifying the Solution

It's crucial to verify our solution to ensure accuracy. We'll substitute the values x = 5 and y = 2 into both original equations to check if they hold true.

Equation 1:

x - y = 3

Substitute x = 5 and y = 2:

5 - 2 = 3

This simplifies to:

3 = 3

The equation holds true.

Equation 2:

-2x + 4y = -2

Substitute x = 5 and y = 2:

-2(5) + 4(2) = -2

This simplifies to:

-10 + 8 = -2

Which further simplifies to:

-2 = -2

This equation also holds true. Since the values x = 5 and y = 2 satisfy both original equations, we have successfully verified our solution.

In conclusion, we've meticulously walked through the process of solving System A, demonstrating each step with clarity and justification. The key steps involved:

  1. Manipulating the equations to create additive inverse coefficients for one variable.
  2. Eliminating a variable by adding the modified equations.
  3. Solving for the remaining variable.
  4. Substituting the value back into an original equation to solve for the other variable.
  5. Verifying the solution by plugging the values into both original equations.

By following these steps, we arrived at the solution (5, 2), which we then verified to be correct. This systematic approach is a cornerstone of solving systems of equations, and mastering it will empower you to tackle a wide array of mathematical problems. Remember, practice is key to solidifying your understanding. Work through various examples, and you'll become increasingly confident in your ability to solve systems of equations effectively.

  • How were the equations in System A manipulated to eliminate a variable?
  • Explain the process of solving for y after eliminating x.
  • What steps are involved in verifying the solution to a system of equations?

Solving Systems of Equations A Step-by-Step Guide with Examples