Solving Systems Of Equations Ordered Pair Solutions

Jul 11, 2025 by ADMIN 52 views

Solving Systems of Equations: A Comprehensive Guide to Finding Ordered Pair Solutions

2x + 5y = 16
-5x - 2y = 2

This article aims to provide a clear and concise explanation, ensuring a thorough understanding of the techniques involved in solving such systems. By the end of this guide, you will be well-equipped to solve similar systems of equations with confidence.

Understanding Systems of Equations

Before diving into the solution, let's first understand what a system of equations represents. A system of equations is a collection of two or more equations with the same set of variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, the solution represents the point(s) where the lines or curves represented by the equations intersect.

In our case, we have two linear equations with two variables, $x$ and $y$ . This means we are looking for a pair of values for $x$ and $y$ that satisfy both equations. There are several methods to solve systems of equations, including substitution, elimination, and graphing. We will focus on the elimination method in this guide.

The Elimination Method: A Step-by-Step Approach

The elimination method involves manipulating the equations in the system to eliminate one of the variables. This is achieved by multiplying one or both equations by a constant so that the coefficients of one of the variables are opposites. When the equations are added together, this variable is eliminated, leaving a single equation with one variable that can be easily solved.

Let's apply the elimination method to our system:

2x + 5y = 16
-5x - 2y = 2

Step 1: Choose a Variable to Eliminate

We can choose to eliminate either $x$ or $y$ . Let's choose to eliminate $x$ . To do this, we need to make the coefficients of $x$ in the two equations opposites. The coefficients of $x$ are 2 and -5. The least common multiple of 2 and 5 is 10. So, we will multiply the first equation by 5 and the second equation by 2 to make the coefficients of $x$ 10 and -10, respectively.

Step 2: Multiply the Equations

Multiply the first equation by 5:

5 * (2x + 5y) = 5 * 16
10x + 25y = 80

Multiply the second equation by 2:

2 * (-5x - 2y) = 2 * 2
-10x - 4y = 4

Now our system looks like this:

10x + 25y = 80
-10x - 4y = 4

Step 3: Add the Equations

Add the two equations together. Notice that the $x$ terms cancel out:

(10x + 25y) + (-10x - 4y) = 80 + 4
10x - 10x + 25y - 4y = 84
21y = 84

Step 4: Solve for the Remaining Variable

Now we have a simple equation with one variable, $y$ . Divide both sides by 21 to solve for $y$ :

21y / 21 = 84 / 21
y = 4

So, we have found that $y = 4$ .

Step 5: Substitute to Find the Other Variable

Substitute the value of $y$ back into either of the original equations to solve for $x$ . Let's use the first original equation:

2x + 5y = 16
2x + 5(4) = 16
2x + 20 = 16

Subtract 20 from both sides:

2x = 16 - 20
2x = -4

Divide both sides by 2:

x = -4 / 2
x = -2

So, we have found that $x = -2$ .

Expressing the Solution as an Ordered Pair

We have found the values of $x$ and $y$ that satisfy the system of equations: $x = -2$ and $y = 4$ . To express the solution as an ordered pair in the format ( $a, b$ ), we write the values of $x$ and $y$ as a pair, with $x$ as the first element and $y$ as the second element. Therefore, the solution to the system is:

(-2, 4)

This ordered pair represents the point where the two lines defined by the equations intersect on a graph.

Verification of the Solution

To ensure our solution is correct, we can substitute the values of $x$ and $y$ back into both original equations and verify that they hold true.

Equation 1:

2x + 5y = 16
2(-2) + 5(4) = 16
-4 + 20 = 16
16 = 16  (True)

Equation 2:

-5x - 2y = 2
-5(-2) - 2(4) = 2
10 - 8 = 2
2 = 2  (True)

Since the values $x = -2$ and $y = 4$ satisfy both equations, our solution is correct.

Alternative Methods for Solving Systems of Equations

While we have focused on the elimination method, it's worth mentioning other methods for solving systems of equations. These include:

Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. This results in a single equation with one variable, which can be solved. The value of this variable is then substituted back into either of the original equations to find the value of the other variable.
Graphing Method: This method involves graphing both equations on the same coordinate plane. The solution to the system is the point(s) where the graphs intersect. This method is particularly useful for visualizing the solution and understanding the relationship between the equations.

The choice of method often depends on the specific system of equations and personal preference. However, the elimination method is generally efficient for systems with linear equations.

Common Mistakes to Avoid

When solving systems of equations, it's important to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

Incorrectly distributing multiplication: Ensure that you multiply every term in the equation by the constant when using the elimination method.
Adding or subtracting equations incorrectly: Pay close attention to the signs of the terms when adding or subtracting equations. A mistake in sign can lead to an incorrect elimination.
Substituting incorrectly: When using the substitution method, ensure you substitute the expression correctly into the other equation. Double-check your work to avoid errors.
Forgetting to solve for both variables: Remember that the solution to a system of equations consists of values for all variables in the system. Don't stop after solving for only one variable.
Not verifying the solution: Always verify your solution by substituting the values back into the original equations to ensure they hold true. This helps catch any errors made during the solution process.

By being mindful of these common mistakes, you can increase your accuracy and confidence in solving systems of equations.

Conclusion

In conclusion, solving systems of equations is a crucial skill in algebra. We have demonstrated the elimination method, a powerful technique for finding solutions. By following the step-by-step approach, we successfully solved the given system:

2x + 5y = 16
-5x - 2y = 2

and expressed the solution as the ordered pair (-2, 4). Remember to verify your solution to ensure accuracy. With practice and a solid understanding of the methods, you can confidently tackle any system of equations that comes your way. Furthermore, we explored alternative methods such as substitution and graphing, as well as common mistakes to avoid, providing a comprehensive guide to mastering this essential mathematical concept. Solving systems of equations has numerous real-world applications, including in fields such as engineering, economics, and computer science. By mastering this skill, you'll be well-equipped to solve a wide range of problems in various disciplines.

Practice Problems

To further solidify your understanding, try solving these practice problems:

Solve the system:

3x - 2y = 7
4x + y = -2

Solve the system:

x + 3y = 10
2x - y = 1

Solve the system:

5x + 2y = 15
-3x + 4y = 7

Remember to express your answers as ordered pairs in the format ( $a, b$ ).

By working through these practice problems, you'll reinforce your understanding of the elimination method and gain confidence in solving systems of equations. Good luck!

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