Solving Systems Of Equations Finding The Correct Ordered Pair

In mathematics, solving systems of equations is a fundamental skill with applications in various fields, from engineering to economics. This article provides a comprehensive guide on how to solve a system of equations and choose the correct ordered pair. We'll walk through the step-by-step process using the given example, ensuring you understand each stage. Mastering this skill will empower you to tackle more complex mathematical problems and real-world scenarios.

Understanding Systems of Equations

A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph. There are several methods to solve systems of equations, including substitution, elimination, and graphing. In this article, we'll focus on the elimination method, which is particularly effective for the given problem.

The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which is easy to solve. Once we find the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable. The solution is then expressed as an ordered pair (x, y), representing the point of intersection of the lines.

To successfully solve a system of equations, it's crucial to understand the underlying principles and apply the appropriate method. The elimination method is particularly useful when the coefficients of one of the variables are multiples of each other or can be easily made multiples by multiplying the equations. Let's dive into solving the given system of equations step by step, ensuring each step is clear and easy to follow.

Problem Statement: Solving the System of Equations

We are given the following system of equations:

3x - 4y = 26
2x + 8y = -36

Our goal is to find the ordered pair (x, y) that satisfies both equations. To do this, we will use the elimination method. The options provided are:

• A. (2, -5)
• B. (2, 5)
• C. (6, 2)
• D. (6, -2)

We need to determine which of these ordered pairs is the correct solution. The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated. This will allow us to solve for the remaining variable and then substitute back to find the other.

Step-by-Step Solution Using the Elimination Method

Step 1: Manipulate the Equations to Eliminate a Variable

Looking at the given equations:

3x - 4y = 26   (Equation 1)
2x + 8y = -36  (Equation 2)

We observe that the coefficients of y are -4 and 8. We can easily eliminate y by multiplying Equation 1 by 2. This will make the coefficient of y in Equation 1 equal to -8, which is the opposite of the coefficient of y in Equation 2.

Multiply Equation 1 by 2:

2 * (3x - 4y) = 2 * 26
6x - 8y = 52   (New Equation 1)

Now we have the modified system:

6x - 8y = 52   (New Equation 1)
2x + 8y = -36  (Equation 2)

Step 2: Add the Equations to Eliminate y

Add the New Equation 1 and Equation 2:

(6x - 8y) + (2x + 8y) = 52 + (-36)
6x - 8y + 2x + 8y = 52 - 36

The y terms cancel out:

8x = 16

Step 3: Solve for x

Divide both sides by 8:

x = 16 / 8
x = 2

Step 4: Substitute the Value of x into One of the Original Equations to Solve for y

We can use either Equation 1 or Equation 2. Let's use Equation 1:

3x - 4y = 26

Substitute x = 2:

3(2) - 4y = 26
6 - 4y = 26

Subtract 6 from both sides:

-4y = 26 - 6
-4y = 20

Divide both sides by -4:

y = 20 / -4
y = -5

Step 5: Write the Solution as an Ordered Pair

The solution is the ordered pair (x, y) = (2, -5).

Verifying the Solution

To ensure our solution is correct, we substitute the values of x and y into both original equations:

For Equation 1:

3x - 4y = 26
3(2) - 4(-5) = 26
6 + 20 = 26
26 = 26  (True)

For Equation 2:

2x + 8y = -36
2(2) + 8(-5) = -36
4 - 40 = -36
-36 = -36  (True)

Since the solution (2, -5) satisfies both equations, it is the correct solution.

Choosing the Correct Ordered Pair

Based on our calculations, the correct ordered pair is (2, -5), which corresponds to option A.

Alternative Methods for Solving Systems of Equations

While we used the elimination method in this example, it's important to be aware of other methods for solving systems of equations. These include:

1. Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. It's particularly useful when one equation is easily solved for one variable in terms of the other.
2. Graphing Method: This method involves graphing both equations on the same coordinate plane. The solution to the system is the point where the lines intersect. This method is useful for visualizing the solution, but it may not be as accurate as algebraic methods.
3. Matrix Method: This method involves using matrices and matrix operations to solve the system of equations. It's particularly useful for larger systems of equations with more variables.

The choice of method often depends on the specific equations in the system and personal preference. However, understanding multiple methods provides flexibility in problem-solving.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Incorrectly Distributing: When multiplying an equation by a constant, make sure to distribute the constant to all terms in the equation.
2. Sign Errors: Pay close attention to the signs of the terms when adding or subtracting equations. A simple sign error can lead to an incorrect solution.
3. Incorrect Substitution: When substituting a value back into an equation, make sure to substitute it correctly and simplify the equation properly.
4. Arithmetic Errors: Double-check your calculations to avoid arithmetic errors. Simple mistakes can lead to incorrect solutions.
5. Not Checking the Solution: Always check your solution by substituting the values back into the original equations. This will help you catch any errors and ensure your solution is correct.

By being aware of these common mistakes and taking the time to double-check your work, you can improve your accuracy and confidence in solving systems of equations.

Real-World Applications of Systems of Equations

Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

1. Economics: Systems of equations are used to model supply and demand, determine equilibrium prices, and analyze market trends.
2. Engineering: Systems of equations are used in structural analysis, circuit analysis, and control systems design.
3. Physics: Systems of equations are used to solve problems in mechanics, electromagnetism, and thermodynamics.
4. Computer Graphics: Systems of equations are used to model transformations, projections, and intersections in computer graphics.
5. Chemistry: Systems of equations are used to balance chemical equations and solve stoichiometry problems.

These are just a few examples of the many ways systems of equations are used in the real world. Understanding how to solve systems of equations is a valuable skill that can be applied in a wide range of fields.

Conclusion

In conclusion, solving systems of equations is a critical mathematical skill with diverse applications. In this article, we demonstrated how to solve the given system of equations using the elimination method. By multiplying the first equation by 2 and adding it to the second equation, we eliminated the variable y and solved for x. Substituting the value of x back into one of the original equations allowed us to solve for y. We found the solution to be the ordered pair (2, -5), which corresponds to option A. This solution was verified by substituting the values back into the original equations.

We also discussed alternative methods for solving systems of equations, common mistakes to avoid, and real-world applications of systems of equations. By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of mathematical problems and real-world challenges. Remember to practice regularly and always double-check your work to ensure accuracy. With consistent effort, you can develop a strong foundation in solving systems of equations and excel in your mathematical endeavors.