Solving 3x - 5y = 21 And X - 4y = 3 A Comprehensive Guide
Introduction to Systems of Linear Equations
In the realm of mathematics, systems of linear equations form a foundational concept with far-reaching applications across various disciplines. These systems, comprising two or more linear equations, describe relationships between variables, and finding solutions involves identifying values that simultaneously satisfy all equations. In this comprehensive exploration, we will delve into the intricacies of solving a specific system of linear equations:
3x - 5y = 21 x - 4y = 3
This system presents a classic example of two linear equations with two unknowns, x and y. Our objective is to determine the unique pair of values for x and y that makes both equations true. This seemingly simple task unlocks a world of mathematical techniques and problem-solving strategies.
Before diving into the solution methods, it's crucial to understand the significance of solving systems of linear equations. These systems arise in diverse fields such as physics, engineering, economics, and computer science. They can model real-world scenarios, enabling us to analyze relationships, make predictions, and optimize outcomes. For instance, in economics, systems of equations can represent supply and demand curves, helping us understand market equilibrium. In engineering, they can model electrical circuits or structural forces. In computer graphics, they are used to perform transformations and render 3D objects. The ability to solve systems of linear equations is thus a fundamental skill for anyone pursuing a career in a quantitative field.
Moreover, the techniques used to solve these systems – substitution, elimination, and matrix methods – are valuable problem-solving tools in themselves. They cultivate logical thinking, analytical skills, and the ability to manipulate equations strategically. These skills extend beyond mathematics, enhancing one's ability to tackle complex problems in any domain. As we embark on this exploration, we will not only find the solution to this specific system but also gain insights into the broader importance of systems of linear equations and the techniques used to solve them.
Methods for Solving Linear Equations
To effectively solve the system of linear equations provided, we can employ several established methods, each with its own strengths and suitability depending on the specific system. The two primary methods we will focus on are the substitution method and the elimination method. These methods are algebraic techniques that systematically manipulate the equations to isolate the variables and ultimately find their values. Understanding both methods provides a versatile toolkit for tackling a wide range of linear systems.
Substitution Method
The substitution 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 a single unknown, which can then be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to determine the value of the other variable. This method is particularly effective when one of the equations can be easily solved for one variable in terms of the other.
Elimination Method
The elimination method, also known as the addition method, involves manipulating the equations such that the coefficients of one of the variables are opposites. By adding the equations together, one variable is eliminated, leaving a single equation with one unknown. This method is particularly useful when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant. Once one variable is found, it can be substituted back into either of the original equations to determine the value of the other variable.
In addition to these algebraic methods, graphical methods can also be used to solve systems of linear equations. By graphing each equation on a coordinate plane, the solution corresponds to the point of intersection of the lines. While graphical methods provide a visual representation of the solution, they may not always yield precise results, especially when the solutions are not integers. However, they can be a valuable tool for understanding the nature of the solutions and verifying algebraic results.
Furthermore, matrix methods, such as Gaussian elimination and Cramer's rule, offer a more systematic approach for solving larger systems of linear equations. These methods are particularly useful in computer applications and engineering problems where systems with dozens or even hundreds of variables are common. While we will not delve into matrix methods in this exploration, it's important to recognize their importance in advanced applications.
In the following sections, we will apply the substitution and elimination methods to solve the given system of equations, illustrating the step-by-step process and highlighting the advantages of each approach.
Solving the System Using Substitution Method
The substitution method provides a straightforward approach to solving the system:
3x - 5y = 21 x - 4y = 3
The essence of this method lies in isolating one variable in one equation and then substituting its equivalent expression into the other equation. This effectively reduces the system to a single equation with a single unknown, which is much easier to solve. Let's apply this method step-by-step to our system.
Step 1: Isolate one variable in one equation.
Looking at the two equations, the second equation, x - 4y = 3, appears to be the easier one to manipulate for isolating a variable. We can readily solve for x by adding 4y to both sides:
x = 4y + 3
Now we have x expressed in terms of y. This expression will be the key to our substitution.
Step 2: Substitute the expression into the other equation.
We now substitute the expression for x (i.e., 4y + 3) into the first equation, 3x - 5y = 21. This replaces x in the first equation with an equivalent expression involving y:
3(4y + 3) - 5y = 21
Notice how the equation now contains only the variable y. This is the crucial step in the substitution method.
Step 3: Solve the resulting equation for the remaining variable.
Next, we simplify and solve the equation for y:
- Distribute the 3: 12y + 9 - 5y = 21
- Combine like terms: 7y + 9 = 21
- Subtract 9 from both sides: 7y = 12
- Divide both sides by 7: y = 12/7
We have now found the value of y: y = 12/7. This is one half of our solution.
Step 4: Substitute the value back into one of the original equations to solve for the other variable.
Now that we know y = 12/7, we can substitute this value back into either of the original equations to find x. It's often easiest to use the equation where we already isolated x, which is x = 4y + 3:
x = 4(12/7) + 3
Simplify to find the value of x:
- Multiply: x = 48/7 + 3
- Find a common denominator: x = 48/7 + 21/7
- Add: x = 69/7
Therefore, x = 69/7. We have now found both x and y.
Step 5: Write the solution as an ordered pair.
The solution to the system is the ordered pair (x, y), which we found to be:
(69/7, 12/7)
This means that the values x = 69/7 and y = 12/7 simultaneously satisfy both equations in the system. We can verify this by substituting these values back into the original equations.
In summary, the substitution method provided a systematic way to solve the system by expressing one variable in terms of the other and then reducing the system to a single equation. This approach highlights the power of algebraic manipulation in solving mathematical problems.
Solving the System Using Elimination Method
The elimination method offers an alternative approach to solving systems of linear equations. This method, also known as the addition method, focuses on strategically manipulating the equations so that when they are added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which can be easily solved. Let's apply the elimination method to the same system:
3x - 5y = 21 x - 4y = 3
Step 1: Multiply one or both equations by a constant to make the coefficients of one variable opposites.
Our goal is to have either the x coefficients or the y coefficients be opposites (e.g., 3 and -3). Looking at the system, it seems easier to make the x coefficients opposites. We can achieve this by multiplying the second equation by -3:
-3(x - 4y) = -3(3)
This gives us the new equation:
-3x + 12y = -9
Now our system looks like this:
3x - 5y = 21 -3x + 12y = -9
Notice that the x coefficients are now opposites (3 and -3).
Step 2: Add the equations together to eliminate one variable.
Now we add the two equations together. The x terms will cancel out:
(3x - 5y) + (-3x + 12y) = 21 + (-9)
Simplifying, we get:
7y = 12
The x variable has been eliminated, leaving us with a single equation in y.
Step 3: Solve the resulting equation for the remaining variable.
We can now easily solve for y:
- Divide both sides by 7: y = 12/7
We have found the value of y: y = 12/7. This is the same value we obtained using the substitution method.
Step 4: Substitute the value back into one of the original equations to solve for the other variable.
As in the substitution method, we substitute the value of y back into one of the original equations to find x. Let's use the second original equation, x - 4y = 3:
x - 4(12/7) = 3
Simplify to find the value of x:
- Multiply: x - 48/7 = 3
- Add 48/7 to both sides: x = 3 + 48/7
- Find a common denominator: x = 21/7 + 48/7
- Add: x = 69/7
Therefore, x = 69/7. This is consistent with the result we obtained using the substitution method.
Step 5: Write the solution as an ordered pair.
The solution to the system is the ordered pair (x, y), which we found to be:
(69/7, 12/7)
This is the same solution we found using the substitution method, confirming the correctness of both approaches.
In summary, the elimination method successfully solved the system by strategically eliminating one variable through addition. This method showcases the power of algebraic manipulation to simplify equations and solve for unknowns. The choice between substitution and elimination often depends on the specific system, with each method offering advantages in different situations. In this case, both methods led to the same solution, highlighting the consistency and reliability of these techniques.
Verification of the Solution
After obtaining a solution to a system of equations, it's crucial to verify its correctness. This step ensures that the values we found for x and y indeed satisfy both equations simultaneously. The verification process involves substituting the obtained values back into the original equations and checking if the equations hold true. This step is not just a formality; it can help identify errors made during the solution process and build confidence in the accuracy of the result. Let's verify our solution, (69/7, 12/7), for the system:
3x - 5y = 21 x - 4y = 3
Verification for the first equation: 3x - 5y = 21
Substitute x = 69/7 and y = 12/7 into the equation:
3(69/7) - 5(12/7) = 21
Simplify the left-hand side:
- Multiply: 207/7 - 60/7 = 21
- Subtract: 147/7 = 21
- Simplify: 21 = 21
The equation holds true. This confirms that our solution satisfies the first equation.
Verification for the second equation: x - 4y = 3
Substitute x = 69/7 and y = 12/7 into the equation:
(69/7) - 4(12/7) = 3
Simplify the left-hand side:
- Multiply: 69/7 - 48/7 = 3
- Subtract: 21/7 = 3
- Simplify: 3 = 3
The equation holds true. This confirms that our solution satisfies the second equation.
Conclusion of Verification
Since the values x = 69/7 and y = 12/7 satisfy both equations in the system, we can confidently conclude that our solution is correct. The verification process not only confirms the accuracy of the solution but also reinforces our understanding of what it means to solve a system of equations. It emphasizes that the solution is a set of values that make all equations in the system true simultaneously.
In the context of mathematics and problem-solving, verification is a crucial step that should never be overlooked. It fosters rigor and precision, essential qualities for success in any quantitative field. By consistently verifying our solutions, we develop a habit of critical thinking and ensure the reliability of our results.
Conclusion
In this comprehensive exploration, we successfully solved the system of linear equations:
3x - 5y = 21 x - 4y = 3
We employed two primary methods – the substitution method and the elimination method – to arrive at the same solution:
(69/7, 12/7)
Both methods demonstrated the power of algebraic manipulation in solving systems of equations. The substitution method involved isolating one variable and substituting its expression into the other equation, while the elimination method focused on strategically eliminating one variable through addition. The choice between these methods often depends on the specific system, with each offering advantages in different scenarios.
Furthermore, we emphasized the importance of verification as a crucial step in the problem-solving process. By substituting our solution back into the original equations, we confirmed its accuracy and gained confidence in our results. Verification not only ensures the correctness of the solution but also reinforces our understanding of the underlying concepts.
Systems of linear equations are a fundamental concept in mathematics with wide-ranging applications across various fields. They provide a framework for modeling relationships between variables and solving real-world problems. The techniques used to solve these systems – substitution, elimination, and matrix methods – are valuable problem-solving tools that cultivate logical thinking and analytical skills.
The solution to a system of linear equations represents the point of intersection of the lines represented by the equations. In the case of two equations with two unknowns, the solution is an ordered pair (x, y) that satisfies both equations simultaneously. Geometrically, this point represents the intersection of the two lines on a coordinate plane.
The ability to solve systems of linear equations is a crucial skill for anyone pursuing a career in a quantitative field. It is essential in areas such as physics, engineering, economics, computer science, and many others. The techniques and concepts learned in solving these systems form a foundation for more advanced mathematical topics and problem-solving approaches.
In conclusion, mastering the art of solving systems of linear equations is not just about finding the right answer; it's about developing a deeper understanding of mathematical relationships, honing problem-solving skills, and cultivating a mindset of rigor and precision. This exploration has provided a detailed guide to solving a specific system, but the principles and techniques discussed are applicable to a wide range of mathematical problems and real-world applications.