Solving Systems Of Equations: A Step-by-Step Guide

Introduction to Solving Systems of Equations

In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. 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 equations in the system true simultaneously. Mastering the techniques to solve these systems is crucial for various applications in science, engineering, economics, and computer science. In this comprehensive guide, we'll break down the process of solving a system of equations, focusing on the specific example:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

This article provides a step-by-step approach to solving this system, ensuring you understand each stage and can apply these techniques to other similar problems. We'll cover the methods of substitution and elimination, offering clear explanations and practical tips to enhance your problem-solving skills. By understanding the underlying principles and practicing consistently, you can confidently tackle any system of equations that comes your way. Let's dive into the world of algebra and unlock the secrets of solving systems of equations!

Understanding the Equations

Before we jump into solving, let's first understand the given system of equations. We have two linear equations:

1. ${3x + y = 11}$
2. ${-2x + y = 1}$

Each equation represents a straight line on a graph. The solution to the system is the point where these two lines intersect. This intersection point provides the values of ${x}$ and ${y}$ that satisfy both equations. To find this point, we can use several methods, but in this case, we'll focus on the elimination method and the substitution method. Understanding the structure of the equations helps us choose the most efficient method to solve them. For instance, noticing that both equations have a ${y}$ term with a coefficient of 1 makes the elimination method particularly straightforward in this scenario. By subtracting one equation from the other, we can eliminate the ${y}$ variable and solve for ${x}$. This initial assessment is crucial for strategic problem-solving. So, with our equations clearly defined, we're ready to move on to the next step and start solving. Remember, the goal is to find the values of ${x}$ and ${y}$ that make both equations true, and understanding the equations is the first step towards achieving that goal.

Method 1: Elimination Method

The elimination method is a powerful technique for solving systems of equations, particularly when the coefficients of one of the variables are the same or easily made the same. In our system:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

We notice that both equations have a ${y}$ term with the same coefficient (1). This makes the elimination method a natural choice. The strategy here is to eliminate one variable by adding or subtracting the equations. Since the ${y}$ terms are identical, we can subtract the second equation from the first to eliminate ${y}$.

Here’s how it works:

Subtract the second equation from the first:

$$(3x + y) - (-2x + y) = 11 - 1$$

This simplifies to:

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

Combining like terms, we get:

$$5x = 10$$

Now, divide both sides by 5 to solve for ${x}$:

$$x = \frac{10}{5} = 2$$

Now that we have the value of ${x}$, we can substitute it into either of the original equations to solve for ${y}$. Let's use the first equation:

$$3x + y = 11$$

Substitute ${x = 2}$:

$$3(2) + y = 11$$

$$6 + y = 11$$

Subtract 6 from both sides:

$$y = 11 - 6 = 5$$

So, the solution to the system of equations is ${x = 2}$ and ${y = 5}$. This means the point of intersection of the two lines represented by these equations is (2, 5). The elimination method is especially effective when you can easily match coefficients, as it streamlines the solving process. By strategically eliminating one variable, we reduce the system to a single equation with one unknown, making it straightforward to solve. This method is a cornerstone of algebra, and mastering it will significantly enhance your equation-solving abilities. Now, let's explore another method to solve the same system and confirm our solution.

Method 2: Substitution Method

The substitution method is another powerful technique for solving systems of equations. This method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved.

Let’s apply the substitution method to our system:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

First, we choose one equation and solve it for one variable. Let’s solve the second equation for ${y}$:

$$-2x + y = 1$$

Add ${2x}$ to both sides:

$$y = 2x + 1$$

Now, we substitute this expression for ${y}$ into the first equation:

$$3x + y = 11$$

Replace ${y}$ with ${2x + 1}$:

$$3x + (2x + 1) = 11$$

Combine like terms:

$$5x + 1 = 11$$

Subtract 1 from both sides:

$$5x = 10$$

Divide both sides by 5:

$$x = \frac{10}{5} = 2$$

Now that we have the value of ${x}$, we can substitute it back into the expression we found for ${y}$:

$$y = 2x + 1$$

Substitute ${x = 2}$:

$$y = 2(2) + 1$$

$$y = 4 + 1 = 5$$

Thus, the solution to the system of equations is ${x = 2}$ and ${y = 5}$, which matches the solution we found using the elimination method. The substitution method is particularly useful when one equation can be easily solved for one variable. By substituting one expression into another, we simplify the system and make it solvable. Both the elimination and substitution methods are valuable tools in algebra, and understanding when to use each can make solving systems of equations much more efficient.

Verification of the Solution

After finding a solution to a system of equations, it's crucial to verify the solution to ensure accuracy. Verification involves substituting the values of ${x}$ and ${y}$ back into the original equations to check if they hold true. This step is essential because it helps catch any errors made during the solving process and ensures that the solution is correct. Let's verify our solution ${x = 2}$ and ${y = 5}$ for the given system:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

First, substitute ${x = 2}$ and ${y = 5}$ into the first equation:

$$3x + y = 11$$

$$3(2) + 5 = 11$$

$$6 + 5 = 11$$

$$11 = 11$$

The first equation holds true. Now, let’s substitute the values into the second equation:

$$-2x + y = 1$$

$$-2(2) + 5 = 1$$

$$-4 + 5 = 1$$

$$1 = 1$$

The second equation also holds true. Since both equations are satisfied by ${x = 2}$ and ${y = 5}$, we can confidently say that this is the correct solution to the system of equations. Verification is a critical step in problem-solving. It not only confirms the accuracy of the solution but also reinforces your understanding of the equations and the solution process. By always verifying your solutions, you develop a habit of precision and ensure that your answers are reliable. This practice is invaluable in mathematics and in any field where problem-solving is essential.

Graphical Interpretation

In addition to algebraic methods, it's beneficial to understand the graphical interpretation of solving systems of equations. Each linear equation in a system can be represented as a straight line on a coordinate plane. The solution to the system is the point where these lines intersect. Visualizing this intersection can provide a deeper understanding of the solution.

Consider our system of equations:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

The first equation, ${3x + y = 11}$, can be rearranged to slope-intercept form (${y = mx + b}$) as:

$$y = -3x + 11$$

This line has a slope of -3 and a y-intercept of 11.

The second equation, ${-2x + y = 1}$, can be rearranged as:

$$y = 2x + 1$$

This line has a slope of 2 and a y-intercept of 1.

When we graph these two lines, we find that they intersect at the point (2, 5). This graphical representation confirms our algebraic solution: ${x = 2}$ and ${y = 5}$. The intersection point visually represents the values that satisfy both equations simultaneously.

Understanding the graphical interpretation adds another dimension to solving systems of equations. It allows you to visualize the solution and reinforces the concept that the solution is the point where the lines meet. This visual understanding can be particularly helpful for students who are visual learners. Moreover, graphical methods can also be used to estimate solutions, especially in cases where algebraic methods are more complex. By connecting algebraic solutions with graphical representations, you gain a more complete and intuitive grasp of systems of equations.

Conclusion

In this guide, we've explored the process of solving the system of equations:

$$\left\{ \begin{aligned} 3x + y &= 11 \\ -2x + y &= 1 \end{aligned} \right.$$

We successfully solved this system using two methods: the elimination method and the substitution method. Both methods led us to the same solution: ${x = 2}$ and ${y = 5}$. We also emphasized the importance of verifying the solution by substituting the values back into the original equations, ensuring the accuracy of our results. Furthermore, we discussed the graphical interpretation of the solution, illustrating how the intersection of the lines represented by the equations corresponds to the solution point (2, 5).

Mastering systems of equations is a fundamental skill in algebra. These techniques are not only crucial for academic success but also have practical applications in various fields, including science, engineering, and economics. By understanding the underlying principles and practicing different methods, you can confidently tackle a wide range of problems. Whether you prefer the elimination method, the substitution method, or a combination of both, the key is to approach each problem systematically and verify your solutions.

Remember, consistent practice is essential for honing your problem-solving skills. The more you work with systems of equations, the more comfortable and proficient you will become. Keep exploring different types of problems and applying the techniques you've learned. With dedication and perseverance, you can master this essential algebraic skill and open doors to more advanced mathematical concepts.

For further learning and practice on systems of equations, you can visit Khan Academy's Systems of Equations Section.