Solving System Of Equations Step By Step Guide

Solving systems of linear equations is a fundamental concept in mathematics, and mastering it is crucial for various applications in science, engineering, and economics. In this article, we will delve into the process of solving a system of linear equations, providing a detailed step-by-step guide to help you understand the underlying principles and techniques. We will use a specific example to illustrate the method, ensuring you grasp the concepts effectively.

The given system of linear equations is:

5x + 9y = -7
2x - 4y = 20

Our goal is to find the values of x and y that satisfy both equations simultaneously. There are several methods to solve such systems, including substitution, elimination, and matrix methods. Here, we will use the elimination method, which is often the most efficient approach for this type of problem.

Step 1: Choose a Variable to Eliminate

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. This allows us to eliminate that variable by adding the equations together. Looking at the given equations, we can choose to eliminate either x or y. Let's choose to eliminate x.

To eliminate x, we need to find a common multiple of the coefficients of x in both equations. The coefficients are 5 and 2, and their least common multiple is 10. We will multiply each equation by a suitable constant so that the coefficients of x become 10 and -10, or vice versa.

Step 2: Multiply the Equations by Suitable Constants

To make the coefficient of x in the first equation 10, we multiply the entire equation by 2:

2 * (5x + 9y) = 2 * (-7)
10x + 18y = -14

To make the coefficient of x in the second equation -10, we multiply the entire equation by -5:

-5 * (2x - 4y) = -5 * (20)
-10x + 20y = -100

Now we have two new equations:

10x + 18y = -14
-10x + 20y = -100

Step 3: Add the Equations to Eliminate a Variable

Now that the coefficients of x are opposites (10 and -10), we can add the two equations together. This will eliminate x and leave us with an equation in terms of y only:

(10x + 18y) + (-10x + 20y) = -14 + (-100)
10x - 10x + 18y + 20y = -114
38y = -114

Step 4: Solve for the Remaining Variable

We now have a simple equation with one variable, y. To solve for y, we divide both sides of the equation by 38:

38y / 38 = -114 / 38
y = -3

So, we have found that y = -3.

Step 5: Substitute the Value Back into One of the Original Equations

To find the value of x, we substitute the value of y we just found (y = -3) into one of the original equations. We can choose either equation; let's use the first equation:

5x + 9y = -7
5x + 9(-3) = -7
5x - 27 = -7

Step 6: Solve for the Other Variable

Now we solve for x. Add 27 to both sides of the equation:

5x - 27 + 27 = -7 + 27
5x = 20

Divide both sides by 5:

5x / 5 = 20 / 5
x = 4

So, we have found that x = 4.

Step 7: Check the Solution

It's always a good idea to check our solution by substituting the values of x and y back into both original equations to make sure they are satisfied. Let's check:

For the first equation:

5x + 9y = -7
5(4) + 9(-3) = -7
20 - 27 = -7
-7 = -7  (Correct)

For the second equation:

2x - 4y = 20
2(4) - 4(-3) = 20
8 + 12 = 20
20 = 20  (Correct)

Since the values x = 4 and y = -3 satisfy both equations, our solution is correct.

Conclusion

The solution to the system of linear equations is x = 4 and y = -3. This corresponds to option C. x = 4, y = -3 in the given choices. The elimination method is a powerful tool for solving systems of linear equations, and by following these steps, you can confidently tackle similar problems. Remember to always check your solution to ensure accuracy.

Understanding Systems of Linear Equations

To further solidify your understanding, let's explore the concept of systems of linear equations in more detail. A system of linear equations is a set of two or more linear equations containing the same variables. The solution to a system of linear equations is the set of values for the variables that make all the equations true simultaneously. Graphically, each linear equation represents a straight line, and the solution to the system is the point where the lines intersect.

There are three possible outcomes when solving a system of two linear equations:

1. Unique Solution: The lines intersect at one point, indicating a single solution for the system. This is the case we encountered in our example. Finding this unique solution is often the goal when solving systems of equations in various mathematical and real-world problems.

2. No Solution: The lines are parallel and do not intersect. In this case, there is no solution that satisfies both equations.

3. Infinitely Many Solutions: The lines are coincident, meaning they are the same line. In this case, every point on the line is a solution to the system.

Methods for Solving Systems of Linear Equations

Besides the elimination method, there are other methods for solving systems of linear equations, each with its strengths and weaknesses:

1. Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved easily. The substitution method is particularly useful when one of the equations is already solved for one variable, or when it's easy to isolate one variable.

2. Elimination Method (Addition/Subtraction Method): As we demonstrated earlier, this method involves manipulating the equations so that the coefficients of one variable are opposites, allowing you to eliminate that variable by adding the equations. The elimination method is often more efficient than substitution when the coefficients of the variables are integers and can be easily manipulated.

3. Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is visually intuitive but may not be accurate for non-integer solutions. The graphical method is best suited for systems with simple equations and when a visual representation of the solution is desired.

4. Matrix Methods: For larger systems of equations (more than two variables), matrix methods such as Gaussian elimination or matrix inversion are often used. These methods are more systematic and can be implemented using computer software. Matrix methods are essential for solving complex systems of linear equations that arise in fields like engineering, physics, and computer science.

Applications of Systems of Linear Equations

Systems of linear equations have numerous applications in various fields:

1. Engineering: Solving circuit problems, structural analysis, and fluid dynamics often involve systems of linear equations. Engineers rely heavily on systems of equations to model and analyze complex systems, ensuring designs are safe and efficient.

2. Economics: Supply and demand models, input-output analysis, and resource allocation problems can be formulated as systems of linear equations. Economists use systems of equations to understand economic behavior and make predictions about market trends.

3. Computer Science: Computer graphics, image processing, and optimization algorithms often use systems of linear equations. In computer graphics, for example, transformations like rotation and scaling can be represented using matrices and solved using linear algebra techniques.

4. Mathematics and Physics: Solving linear differential equations, finding equilibrium points in dynamical systems, and analyzing vector spaces all involve systems of linear equations. The fundamental laws of physics are often expressed as mathematical equations, and solving these equations frequently requires solving systems of linear equations.

5. Real-World Problems: Mixture problems, distance-rate-time problems, and investment problems can often be modeled and solved using systems of linear equations. By translating real-world scenarios into mathematical equations, we can use the tools of linear algebra to find solutions and make informed decisions.

Tips for Solving Systems of Linear Equations

Here are some helpful tips to keep in mind when solving systems of linear equations:

1. Choose the Method Wisely: Select the method that seems most efficient for the given system. Substitution may be better if one equation is already solved for a variable, while elimination may be more suitable if the coefficients are easily manipulated.

2. Be Organized: Keep your work neat and organized to avoid errors. Write down each step clearly and label your equations.

3. Check Your Solution: Always check your solution by substituting the values back into the original equations. This will help you catch any mistakes.

4. Look for Special Cases: Be aware of cases where there is no solution (parallel lines) or infinitely many solutions (coincident lines).

5. Practice Regularly: The more you practice, the more comfortable you will become with solving systems of linear equations. Work through a variety of examples to develop your skills.

By mastering the concepts and techniques discussed in this article, you will be well-equipped to solve systems of linear equations and apply them to various real-world problems. Remember, the key is to understand the underlying principles, choose the appropriate method, and practice consistently.