Solving Systems Of Equations A Comprehensive Guide

Solving systems of equations is a fundamental skill in mathematics, with applications spanning various fields, from science and engineering to economics and computer science. In this comprehensive guide, we'll delve into the methods for tackling these systems, focusing on the specific example you provided. But first, let's understand what a system of equations really is.

A system of equations, at its core, is a collection of two or more equations that share variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Think of it as finding the sweet spot where all the equations agree. These equations can represent lines, curves, or even more complex relationships, and the solutions represent the points where these relationships intersect.

Understanding the Problem: A Linear System

The system of equations you presented is a classic example of a linear system:

$\begin{array}{l} x-3 y=5 \\ 2 x+y=10 \end{array}$

This system consists of two linear equations, each representing a straight line on a graph. The solution to this system will be the point (x, y) where these two lines intersect. There are several methods we can use to find this point, and we'll explore two of the most common ones: substitution and elimination.

Method 1: The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable that we can easily solve. Let's apply this to our system.

Step 1: Solve one equation for one variable.

Let's take the first equation, x - 3y = 5, and solve it for x. Adding 3y to both sides, we get:

x = 3y + 5

Step 2: Substitute the expression into the other equation.

Now, we'll substitute this expression for x (which is 3y + 5) into the second equation, 2x + y = 10:

2(3y + 5) + y = 10

Step 3: Solve for the remaining variable.

Now we have an equation with only y. Let's simplify and solve for y:

6y + 10 + y = 10

7y + 10 = 10

7y = 0

y = 0

Step 4: Substitute the value back to find the other variable.

Now that we know y = 0, we can substitute this value back into either of the original equations or the expression we found for x. Let's use the expression x = 3y + 5:

x = 3(0) + 5

x = 5

Therefore, using the substitution method, we've found the solution to be x = 5 and y = 0, which corresponds to the point (5, 0). This means the lines intersect at the point (5,0).

Method 2: The Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one variable are opposites. When we add the equations together, this variable is eliminated, leaving us with a single equation in one variable. Let's see how this works for our system.

Step 1: Manipulate the equations to match the coefficients.

Looking at our system:

$\begin{array}{l} x-3 y=5 \\ 2 x+y=10 \end{array}$

Notice that the coefficient of y in the first equation is -3, and in the second equation, it's 1. To eliminate y, we can multiply the second equation by 3:

3(2x + y) = 3(10)

6x + 3y = 30

Now our system looks like this:

$\begin{array}{l} x-3 y=5 \\ 6x+3y=30 \end{array}$

Step 2: Add the equations together.

Now, we can add the two equations together. Notice that the y terms will cancel out:

(x - 3y) + (6x + 3y) = 5 + 30

7x = 35

Step 3: Solve for the remaining variable.

Solving for x, we get:

x = 35 / 7

x = 5

Step 4: Substitute the value back to find the other variable.

Now that we know x = 5, we can substitute this value back into either of the original equations. Let's use the first equation, x - 3y = 5:

5 - 3y = 5

-3y = 0

y = 0

Again, using the elimination method, we've found the solution to be x = 5 and y = 0, which corresponds to the point (5, 0). This confirms our result from the substitution method, meaning the intersection point remains (5,0).

Verifying the Solution

It's always a good idea to verify our solution by plugging the values of x and y back into the original equations to make sure they hold true.

For the first equation, x - 3y = 5:

5 - 3(0) = 5

5 = 5 (This is true)

For the second equation, 2x + y = 10:

2(5) + 0 = 10

10 = 10 (This is also true)

Since our solution satisfies both equations, we can be confident that (5, 0) is indeed the correct solution.

The Answer

Therefore, the solution to the system of equations is (5, 0), which corresponds to option A. Guys, we nailed it!

Why Understanding Systems of Equations Matters

Solving systems of equations might seem like an abstract mathematical exercise, but it has incredibly practical applications in the real world. Here's why it's a crucial skill to master:

• Modeling Real-World Problems: Many real-world situations can be modeled using systems of equations. For example, in economics, we can model supply and demand curves as a system of equations, and the solution represents the equilibrium price and quantity. In physics, we can use systems of equations to analyze the motion of objects or the flow of electricity in a circuit. Systems of equations allow us to formulate mathematical models that accurately describe real-world phenomena.
• Optimization: Systems of equations are often used in optimization problems, where we want to find the best solution given certain constraints. For example, a business might use a system of equations to determine the optimal production levels to maximize profit, considering factors like production costs, market demand, and resource availability. Businesses can make informed decisions by leveraging systems of equations for optimization.
• Computer Graphics and Game Development: In computer graphics and game development, systems of equations are used extensively to perform transformations, such as rotations, scaling, and translations. These transformations are essential for creating realistic and visually appealing graphics. Game developers rely on systems of equations to simulate complex interactions and movements.
• Data Analysis and Machine Learning: Systems of equations are used in various data analysis techniques, such as linear regression, which is used to find the best-fit line for a set of data points. This is a fundamental concept in machine learning, where we use data to train models that can make predictions or classifications. Systems of equations play a crucial role in training machine learning algorithms.

Tips and Tricks for Solving Systems of Equations

Now that we've covered the main methods and the importance of solving systems of equations, let's talk about some tips and tricks that can make the process even smoother:

• Choose the Best Method: Both the substitution and elimination methods are powerful tools, but one might be more efficient than the other depending on the specific system. If one equation is already solved for a variable or has a simple coefficient, substitution might be easier. If the coefficients of one variable are easily made opposites, elimination might be the better choice. Consider the structure of the equations to select the most appropriate method.
• Look for Simplifications: Before diving into the substitution or elimination process, take a moment to see if you can simplify the equations. This might involve combining like terms, dividing both sides of an equation by a common factor, or rearranging terms. Simplifying the equations can make the subsequent steps much easier. Simplification can reduce the chances of errors and streamline the solution process.
• Be Organized: Solving systems of equations can involve multiple steps, so it's essential to stay organized. Write down each step clearly, and double-check your work as you go. Use a consistent notation, and keep track of which equations you're working with. Maintaining organization throughout the process can prevent mistakes and confusion. Keeping track of each step ensures accuracy in the final solution.
• Check Your Solution: As we demonstrated earlier, always check your solution by plugging the values back into the original equations. This is the best way to catch any errors you might have made along the way. If the solution doesn't satisfy both equations, you know you need to go back and review your work. Verification is a crucial step in ensuring the correctness of the solution.
• Practice Makes Perfect: Like any mathematical skill, solving systems of equations becomes easier with practice. Work through a variety of examples, and don't be afraid to make mistakes. Each mistake is a learning opportunity. The more you practice, the more comfortable and confident you'll become in solving these systems. Regular practice solidifies your understanding and problem-solving abilities.

Beyond Two Variables

While we've focused on systems of two equations with two variables, the concepts extend to larger systems. For example, you might encounter systems of three equations with three variables, or even more. The same methods, substitution and elimination, can be applied, but the process can become more complex. Matrix methods, such as Gaussian elimination, are often used to solve larger systems efficiently. These advanced techniques provide streamlined approaches for handling complex systems.

Conclusion

Solving systems of equations is a valuable skill with wide-ranging applications. By mastering the substitution and elimination methods, and by understanding the underlying concepts, you'll be well-equipped to tackle a variety of mathematical problems and real-world scenarios. Remember to stay organized, practice regularly, and always check your work. With a solid foundation in solving systems of equations, you'll be prepared for more advanced mathematical concepts and their applications in various fields. Keep practicing, guys, and you'll become equation-solving pros in no time!