Solving Systems Of Linear Equations A Step-by-Step Guide

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In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. Solving these systems means finding values for the variables that satisfy all equations simultaneously. This article will guide you through the process of solving a specific system of linear equations, providing a detailed explanation and step-by-step instructions to help you understand the underlying concepts and techniques. We will focus on the following system:

-2x + 4y = 16
2x + 2y = 8

Our goal is to find the ordered pair (x, y) that satisfies both equations. We'll explore a common method for solving such systems: the elimination method.

Understanding the Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations by strategically eliminating one variable, making it possible to solve for the remaining variable. The key idea behind this method is to manipulate the equations so that the coefficients of one of the variables are opposites (e.g., 2 and -2). When the equations are added together, this variable is eliminated, leaving an equation with only one variable that can be easily solved. Once we find the value of one variable, we can substitute it back into either of the original equations to solve for the other variable.

The elimination method is particularly useful when the equations are in standard form (Ax + By = C), but it can also be adapted for other forms. The steps involved in the elimination method are as follows:

  1. Align the Equations: Write the equations one above the other, aligning the variables (x and y) and the constants.
  2. Create Opposing Coefficients: Multiply one or both equations by a constant so that the coefficients of either x or y are opposites (same number, opposite signs).
  3. Eliminate a Variable: Add the equations together. The variable with opposing coefficients will cancel out, leaving an equation with only one variable.
  4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
  5. Substitute and Solve: Substitute the value found in step 4 into either of the original equations and solve for the other variable.
  6. Check the Solution: Substitute both values (x and y) into both original equations to verify that the solution satisfies both equations.

The elimination method is a versatile and reliable technique for solving systems of linear equations, and it's a valuable tool in algebra and beyond.

Step-by-Step Solution

Step 1: Align the Equations

First, we align the equations as they are already presented:

-2x + 4y = 16
2x + 2y = 8

Notice that the x terms and the y terms are aligned, making it easier to proceed with the elimination method. Alignment is a crucial first step in ensuring clarity and organization when solving systems of equations.

Step 2: Create Opposing Coefficients

Observe that the coefficients of x in the two equations are already opposites (-2 and 2). This is ideal for the elimination method, as we can proceed directly to the next step without needing to multiply either equation by a constant. Recognizing these pre-existing opposing coefficients saves us a step and simplifies the solution process. In cases where coefficients are not opposites, we would need to multiply one or both equations by appropriate constants to create them.

Step 3: Eliminate a Variable

Now, we add the two equations together:

(-2x + 4y) + (2x + 2y) = 16 + 8

Combining like terms, we get:

-2x + 2x + 4y + 2y = 24
0x + 6y = 24
6y = 24

As you can see, the x terms have been eliminated, leaving us with a simple equation involving only y. This is the core of the elimination method – strategically removing one variable to solve for the other. By adding the equations, we effectively canceled out the x variable, allowing us to isolate and solve for y.

Step 4: Solve for the Remaining Variable

To solve for y, we divide both sides of the equation 6y = 24 by 6:

6y / 6 = 24 / 6
y = 4

Thus, we have found the value of y to be 4. This is a significant step towards finding the complete solution to the system of equations. The value of y will now be used to find the corresponding value of x.

Step 5: Substitute and Solve

Substitute y = 4 into either of the original equations. Let's use the second equation:

2x + 2y = 8
2x + 2(4) = 8
2x + 8 = 8

Now, subtract 8 from both sides:

2x = 0

Divide both sides by 2:

x = 0

Therefore, we find that x = 0. We now have both x and y values, giving us a potential solution to the system of equations.

Step 6: Check the Solution

To verify our solution, we substitute x = 0 and y = 4 into both original equations:

Equation 1:

-2x + 4y = 16
-2(0) + 4(4) = 16
0 + 16 = 16
16 = 16 (True)

Equation 2:

2x + 2y = 8
2(0) + 2(4) = 8
0 + 8 = 8
8 = 8 (True)

Since the solution (0, 4) satisfies both equations, it is the correct solution to the system.

The Solution and Its Implications

The ordered pair that solves the system of equations is (0, 4). This means that the point (0, 4) is the intersection of the two lines represented by the equations -2x + 4y = 16 and 2x + 2y = 8 on a coordinate plane. Graphically, this is the single point where the two lines meet. The solution (0, 4) is unique for this system of equations, indicating that the lines are independent and intersect at exactly one point.

Understanding the solution to a system of linear equations has practical applications in various fields. For example, in economics, it can represent the equilibrium point where supply and demand curves intersect. In engineering, it can be used to solve for unknown variables in circuit analysis or structural mechanics. The ability to solve systems of equations is a fundamental skill in mathematics and has broad applicability in real-world problems.

Alternative Methods for Solving Linear Systems

While the elimination method is effective for this particular system, it's important to be aware of other methods for solving systems of linear equations. Two common alternatives are:

  1. Substitution Method: In the substitution method, one equation is solved for one variable in terms of the other variable. This expression is then substituted into the other equation, resulting in an equation with only one variable. This method is particularly useful when one equation is already solved for one variable or can be easily solved.
  2. Graphical Method: The graphical method involves plotting the lines represented by the equations on a coordinate plane. The solution to the system is the point where the lines intersect. This method is visually intuitive but may not be as precise as algebraic methods, especially when the solution involves fractions or decimals. However, it provides a valuable visual representation of the system and its solution.

The choice of method depends on the specific system of equations and personal preference. For some systems, one method may be more efficient than others. For instance, the elimination method worked well in this case because the coefficients of x were already opposites. However, for other systems, the substitution method might be more straightforward.

Potential Pitfalls and Common Mistakes

When solving systems of linear equations, it's essential to be mindful of potential pitfalls and common mistakes. These can lead to incorrect solutions and wasted effort. Here are some common errors to watch out for:

  1. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition, subtraction, multiplication, or division, can easily throw off the entire solution. It's crucial to double-check each step and use a calculator if needed.
  2. Sign Errors: Incorrectly handling negative signs is a frequent source of errors. Pay close attention to signs when adding, subtracting, and multiplying equations. A single sign error can lead to an incorrect value for one or both variables.
  3. Incorrect Substitution: When using the substitution method, ensure that you substitute the expression correctly into the other equation. Substituting into the same equation you solved for the variable will not lead to a solution.
  4. Forgetting to Check the Solution: Always check your solution by substituting the values of x and y into both original equations. This step is crucial for verifying that your solution satisfies the system and for catching any errors made during the solving process.
  5. Misinterpreting the Result: If the lines are parallel, there is no solution to the system. If the lines are the same, there are infinitely many solutions. Make sure to interpret the result in the context of the problem.

By being aware of these potential pitfalls and common mistakes, you can increase your accuracy and confidence when solving systems of linear equations.

Conclusion

In conclusion, we have successfully solved the system of linear equations:

-2x + 4y = 16
2x + 2y = 8

Using the elimination method, we found the solution to be the ordered pair (0, 4). This solution satisfies both equations, representing the point of intersection of the two lines on a coordinate plane. We explored the step-by-step process of the elimination method, including aligning the equations, creating opposing coefficients, eliminating a variable, solving for the remaining variable, substituting to find the other variable, and checking the solution.

Furthermore, we discussed alternative methods for solving linear systems, such as the substitution and graphical methods, highlighting their strengths and when they might be preferred. We also addressed potential pitfalls and common mistakes to avoid when solving these types of problems. Understanding these concepts and techniques is crucial for mastering algebra and its applications in various fields. Solving systems of linear equations is a fundamental skill that empowers you to tackle real-world problems in mathematics, science, engineering, economics, and more.

Therefore, the correct answer is A. (0, 4). This comprehensive guide should equip you with the knowledge and skills to confidently solve similar systems of linear equations in the future.