Solving Systems Of Linear Equations Finding The Correct Ordered Pair

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When you are presented with a system of linear equations, the primary goal is to find the values of the variables that satisfy all equations simultaneously. In simpler terms, we are looking for the point where the lines represented by these equations intersect on a graph. This point of intersection is the solution to the system, and it is typically expressed as an ordered pair (x, y). In this article, we will explore a detailed approach to determining which ordered pair, from a given set of options, is a solution to the system.

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same variables. A linear equation is an equation that can be written in the form Ax + By = C, where A, B, and C are constants, and x and y are variables. The solution to a system of linear equations is the set of values for the variables that make all the equations true.

To solve a system of two linear equations with two variables, we need to find the ordered pair (x, y) that satisfies both equations simultaneously. There are several methods to solve such systems, including graphing, substitution, and elimination. However, when presented with a multiple-choice question like the one above, the most straightforward approach is often to test each ordered pair to see if it satisfies both equations.

The Given System of Equations

Let's consider the system of linear equations provided:

2x + 3y = 6
-3x + 5y = 10

Our task is to determine which of the following ordered pairs is a solution to this system:

  • A. (0, 2)
  • B. (2, 0)
  • C. (3, 2)
  • D. (2, 3)

Step-by-Step Verification Process

The most efficient way to solve this problem is to substitute the x and y values from each ordered pair into both equations and check if the equations hold true. If an ordered pair satisfies both equations, it is a solution to the system.

A. Checking the Ordered Pair (0, 2)

Let's substitute x = 0 and y = 2 into the first equation:

2(0) + 3(2) = 6
0 + 6 = 6
6 = 6

The first equation holds true. Now, let's substitute x = 0 and y = 2 into the second equation:

-3(0) + 5(2) = 10
0 + 10 = 10
10 = 10

The second equation also holds true. Since (0, 2) satisfies both equations, it is a solution to the system.

B. Checking the Ordered Pair (2, 0)

Let's substitute x = 2 and y = 0 into the first equation:

2(2) + 3(0) = 6
4 + 0 = 6
4 = 6

The first equation does not hold true. Since (2, 0) does not satisfy the first equation, it cannot be a solution to the system. We do not need to check the second equation.

C. Checking the Ordered Pair (3, 2)

Let's substitute x = 3 and y = 2 into the first equation:

2(3) + 3(2) = 6
6 + 6 = 6
12 = 6

The first equation does not hold true. Since (3, 2) does not satisfy the first equation, it cannot be a solution to the system.

D. Checking the Ordered Pair (2, 3)

Let's substitute x = 2 and y = 3 into the first equation:

2(2) + 3(3) = 6
4 + 9 = 6
13 = 6

The first equation does not hold true. Since (2, 3) does not satisfy the first equation, it cannot be a solution to the system.

Conclusion and Final Answer

After systematically checking each ordered pair, we found that only the ordered pair (0, 2) satisfies both equations in the system. Therefore, the solution to the system of linear equations is (0, 2).

Final Answer: The correct answer is A. (0, 2).

Importance of Verifying Solutions

In solving systems of equations, it is crucial to verify your solutions by substituting them back into the original equations. This step ensures that the values you have found are indeed correct and satisfy all the given conditions. By verifying, you can avoid errors and gain confidence in your answer. This practice is especially important in more complex problems where mistakes can easily occur.

Alternative Methods for Solving Systems of Equations

While substituting ordered pairs is an efficient method for multiple-choice questions, it's important to be aware of other techniques for solving systems of linear equations. These methods include:

  1. Graphing: This method involves plotting the two lines on a coordinate plane. The point of intersection represents the solution to the system. Graphing is a visual method that can be helpful for understanding the concept of solutions, but it may not always provide precise answers if the intersection point has non-integer coordinates.

  2. Substitution: In the substitution method, you solve one equation for one variable and substitute that expression into the other equation. This results in a single equation with one variable, which can then be easily solved. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.

  3. Elimination (or Addition): The elimination method involves adding or subtracting the equations in such a way that one of the variables is eliminated. This is achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When you add the equations, that variable is eliminated, and you can solve for the remaining variable. The elimination method is effective when the coefficients of one variable are easily made opposites.

Real-World Applications of Systems of Linear Equations

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

  • Economics: In economics, systems of equations can be used to model supply and demand curves. The solution to the system represents the equilibrium price and quantity, where supply equals demand.
  • Engineering: Engineers use systems of equations to analyze circuits, design structures, and solve problems in fluid mechanics. For example, they might use systems of equations to determine the forces acting on a bridge or the flow rates in a network of pipes.
  • Computer Graphics: Systems of equations are used in computer graphics to perform transformations such as scaling, rotation, and translation of objects in a 3D scene. These transformations are essential for creating realistic images and animations.
  • Mixture Problems: Many problems involving mixtures of different substances can be solved using systems of equations. For example, if you need to mix two solutions with different concentrations to obtain a desired concentration, you can set up a system of equations to determine the required amounts of each solution.
  • Navigation: Systems of equations play a role in GPS (Global Positioning System) technology. The GPS receiver uses signals from multiple satellites to determine its location. This involves solving a system of equations to find the coordinates of the receiver.

Tips for Solving Systems of Linear Equations

  • Read the problem carefully: Understand what the problem is asking and identify the variables and equations involved.
  • Choose the appropriate method: Depending on the problem, one method (substitution, elimination, or graphing) may be more efficient than others.
  • Keep your work organized: Write each step clearly and neatly to avoid mistakes.
  • Check your solution: Substitute your solution back into the original equations to verify that it is correct.
  • Practice regularly: The more you practice, the better you will become at solving systems of equations.

Conclusion

In conclusion, determining whether an ordered pair is a solution to a system of linear equations involves substituting the values into each equation and verifying that they hold true. This method is particularly useful for multiple-choice questions. However, it's also important to understand other methods for solving systems of equations, such as graphing, substitution, and elimination, as they provide a broader toolkit for tackling different types of problems. Systems of linear equations have wide-ranging applications in various fields, making it a fundamental topic in mathematics and its practical applications. By following a systematic approach and practicing regularly, you can master the skills needed to solve these problems effectively.