Solving 3x - 2y = 10 And X + 3y = 7 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving simultaneous equations is a fundamental skill with widespread applications in various fields, including engineering, economics, and computer science. Simultaneous equations, also known as systems of equations, involve finding the values of two or more variables that satisfy a set of equations simultaneously. This article delves into the step-by-step process of solving the given system of equations: 3x - 2y = 10 and x + 3y = 7. We will explore different methods, including substitution and elimination, to arrive at the solution. Understanding how to solve simultaneous equations is crucial for tackling more complex mathematical problems and real-world scenarios where multiple variables and constraints are involved.
Understanding Simultaneous Equations
Before we dive into the solution, let's first understand what simultaneous equations are and why they are important. Simultaneous equations are a set of two or more equations containing two or more variables where we seek values for the variables that satisfy all equations concurrently. In simpler terms, we're looking for a set of numbers that, when plugged into the variables, make all the equations true at the same time. The given system, 3x - 2y = 10 and x + 3y = 7, is a classic example of a system of two linear equations with two variables, x and y. These equations represent straight lines when graphed, and the solution to the system corresponds to the point where the lines intersect. The ability to solve such systems is not just a mathematical exercise; it's a powerful tool for modeling and solving real-world problems. For instance, in economics, we might use simultaneous equations to determine the equilibrium price and quantity in a market, given supply and demand equations. In engineering, they can be used to analyze the forces acting on a structure. The importance of mastering this skill cannot be overstated, as it forms the basis for more advanced mathematical concepts and applications.
Method 1: Solving by Substitution
The substitution method is a powerful technique for solving simultaneous equations. The core idea behind this method is to solve one equation for one variable and then substitute that expression into the other equation. This process eliminates one variable, leaving us with a single equation in one variable, which we can easily solve. Let's apply this method to our system of equations: 3x - 2y = 10 and x + 3y = 7. The first step is to choose one equation and solve for one variable. Looking at the equations, it seems easiest to solve the second equation (x + 3y = 7) for x. Subtracting 3y from both sides, we get x = 7 - 3y. Now, we substitute this expression for x into the first equation (3x - 2y = 10). Replacing x with (7 - 3y), we get 3(7 - 3y) - 2y = 10. This equation now contains only one variable, y. We proceed to simplify and solve for y: 21 - 9y - 2y = 10, which simplifies to 21 - 11y = 10. Subtracting 21 from both sides gives -11y = -11, and dividing by -11, we find y = 1. Now that we have the value of y, we can substitute it back into either of the original equations to find x. Using the equation x = 7 - 3y, we substitute y = 1 to get x = 7 - 3(1) = 7 - 3 = 4. Therefore, the solution to the system of equations using the substitution method is x = 4 and y = 1. This means that the point (4, 1) is the intersection of the two lines represented by the equations.
Step-by-Step Breakdown of Substitution
To further clarify the substitution method, let's break down the process into detailed steps. This will help solidify your understanding and make it easier to apply this method to other systems of equations. Step 1: Choose an Equation and Solve for a Variable. Look at the given equations (3x - 2y = 10 and x + 3y = 7) and decide which equation is easier to solve for one of the variables. In this case, the second equation (x + 3y = 7) is simpler to solve for x. Isolate x by subtracting 3y from both sides: x = 7 - 3y. This is our expression for x in terms of y. Step 2: Substitute the Expression into the Other Equation. Now, take the expression we found for x (x = 7 - 3y) and substitute it into the other equation (3x - 2y = 10). This is a crucial step as it eliminates one variable. Replacing x with (7 - 3y), we get 3(7 - 3y) - 2y = 10. This equation now only contains the variable y. Step 3: Simplify and Solve for the Remaining Variable. Simplify the equation obtained in Step 2. Distribute the 3 in the expression 3(7 - 3y) to get 21 - 9y. So, the equation becomes 21 - 9y - 2y = 10. Combine like terms: 21 - 11y = 10. Subtract 21 from both sides: -11y = -11. Divide both sides by -11 to solve for y: y = 1. We have now found the value of y. Step 4: Substitute the Value Back to Find the Other Variable. Substitute the value of y we found (y = 1) back into the expression we obtained in Step 1 (x = 7 - 3y) to find the value of x. Replace y with 1: x = 7 - 3(1) = 7 - 3 = 4. So, x = 4. Step 5: Verify the Solution. It's always a good idea to verify the solution by substituting the values of x and y back into both original equations to ensure they hold true. For the first equation (3x - 2y = 10): 3(4) - 2(1) = 12 - 2 = 10, which is true. For the second equation (x + 3y = 7): 4 + 3(1) = 4 + 3 = 7, which is also true. Since the values x = 4 and y = 1 satisfy both equations, we have successfully solved the system using the substitution method. This step-by-step breakdown provides a clear and concise guide to applying the substitution method to solve simultaneous equations. Practice with different examples will further enhance your proficiency.
Method 2: Solving by Elimination
The elimination method, also known as the addition or subtraction method, provides another effective strategy for solving systems of equations. The main concept behind the elimination method is to manipulate the equations in such a way that when they are added or subtracted, one of the variables is eliminated. This leaves us with a single equation in one variable, which can then be solved easily. Let's apply this method to the given system of equations: 3x - 2y = 10 and x + 3y = 7. The goal is to make the coefficients of either x or y the same (or additive inverses) in both equations. To eliminate x, we can multiply the second equation by -3, which will give us a -3x term. This gives us the modified system: 3x - 2y = 10 and -3x - 9y = -21. Now, we add the two equations together. The 3x and -3x terms cancel each other out, eliminating x: (3x - 2y) + (-3x - 9y) = 10 + (-21). This simplifies to -11y = -11. Dividing both sides by -11, we find y = 1. Now that we have the value of y, we can substitute it back into either of the original equations to find x. Let's use the second equation, x + 3y = 7. Substituting y = 1, we get x + 3(1) = 7, which simplifies to x + 3 = 7. Subtracting 3 from both sides gives x = 4. Thus, the solution to the system of equations using the elimination method is x = 4 and y = 1, consistent with the result obtained using the substitution method. The elimination method is particularly useful when the equations are structured in a way that makes it easy to eliminate a variable by addition or subtraction.
Detailed Steps for the Elimination Method
To ensure a comprehensive understanding of the elimination method, let's break down the process into a series of clear, manageable steps. This detailed breakdown will help you confidently apply the method to various systems of equations. Step 1: Align the Equations and Choose a Variable to Eliminate. First, make sure the equations are aligned, with like terms (x terms, y terms, and constants) in the same columns. Our equations are already in this format: 3x - 2y = 10 and x + 3y = 7. Next, choose which variable you want to eliminate. Look for the variable whose coefficients are easiest to make the same (or additive inverses). In this case, it might seem simpler to eliminate x because we can easily multiply the second equation by -3 to get a -3x term. Step 2: Multiply One or Both Equations to Make the Coefficients Match or Add Up to Zero. We want to make the coefficients of x in both equations additive inverses. To do this, multiply the second equation (x + 3y = 7) by -3. This gives us -3(x + 3y) = -3(7), which simplifies to -3x - 9y = -21. Now we have the modified system: 3x - 2y = 10 and -3x - 9y = -21. Notice that the coefficients of x are now 3 and -3, which are additive inverses. Step 3: Add or Subtract the Equations to Eliminate the Chosen Variable. Add the two equations together: (3x - 2y) + (-3x - 9y) = 10 + (-21). This simplifies to 3x - 2y - 3x - 9y = 10 - 21. Combine like terms: -11y = -11. The x terms have been eliminated, leaving us with an equation in just y. Step 4: Solve for the Remaining Variable. Solve the equation -11y = -11 for y. Divide both sides by -11: y = 1. We have now found the value of y. Step 5: Substitute the Value Back into One of the Original Equations to Find the Other Variable. Substitute the value of y (y = 1) back into either of the original equations to find x. Let's use the second equation, x + 3y = 7. Substitute y = 1: x + 3(1) = 7. Simplify: x + 3 = 7. Subtract 3 from both sides: x = 4. So, x = 4. Step 6: Verify the Solution. As with the substitution method, it's crucial to verify the solution. Substitute x = 4 and y = 1 back into both original equations. For the first equation (3x - 2y = 10): 3(4) - 2(1) = 12 - 2 = 10, which is true. For the second equation (x + 3y = 7): 4 + 3(1) = 4 + 3 = 7, which is also true. Since the values satisfy both equations, the solution is correct. This detailed step-by-step guide to the elimination method should provide a clear understanding of the process. Practice with various examples to master this technique and enhance your problem-solving skills.
Comparing Substitution and Elimination
Both substitution and elimination are powerful methods for solving simultaneous equations, but they have their strengths and weaknesses. Choosing the right method can often simplify the solving process. The substitution method shines when one of the equations can be easily solved for one variable in terms of the other. In such cases, like our example where x + 3y = 7 can be easily rearranged to x = 7 - 3y, substitution becomes a straightforward choice. It involves replacing one variable in the other equation, reducing the system to a single equation with one variable. This method is particularly advantageous when dealing with systems where isolating a variable is relatively simple. On the other hand, the elimination method excels when the coefficients of one of the variables are either the same or additive inverses, or can be made so with a simple multiplication. This method involves adding or subtracting the equations to eliminate one variable, which is especially efficient when the equations are structured in a way that allows for easy cancellation. For instance, if we had equations like 2x + 3y = 5 and 4x - 3y = 1, elimination would be a natural choice because the y terms have coefficients that are additive inverses. In general, the best approach is to assess the given system of equations and choose the method that seems most efficient. Sometimes, a combination of both methods might be the most effective strategy. Regardless of the method chosen, the key is to perform the steps carefully and systematically to avoid errors. Verifying the solution by substituting the values back into the original equations is always a crucial step to ensure accuracy.
Real-World Applications
Simultaneous equations are not just abstract mathematical concepts; they are powerful tools with numerous real-world applications. These systems of equations allow us to model and solve problems in various fields, ranging from science and engineering to economics and everyday situations. In physics, simultaneous equations can be used to analyze the motion of objects under multiple forces or to calculate the currents and voltages in electrical circuits. For example, Kirchhoff's laws in circuit analysis often lead to a system of linear equations that need to be solved simultaneously to determine the currents in different branches of the circuit. In chemistry, they can be used to balance chemical equations, where the number of atoms of each element must be the same on both sides of the equation. This often results in a system of equations that needs to be solved to find the stoichiometric coefficients. In economics, supply and demand models are often represented by simultaneous equations. The equilibrium price and quantity in a market can be found by solving the system of equations representing the supply and demand curves. Linear programming, a technique used to optimize resource allocation, also relies heavily on solving systems of linear equations and inequalities. In engineering, simultaneous equations are used in structural analysis to determine the forces and stresses in different parts of a structure. They are also used in control systems to design controllers that meet specific performance requirements. Even in everyday life, we encounter situations where solving simultaneous equations can be helpful. For example, if you are trying to determine the cost of two different items given their combined price and the price difference, you can set up a system of equations and solve it. Consider this scenario: Suppose you buy 2 apples and 3 bananas for $5, and you also know that apples cost $0.50 more than bananas. You can set up a system of equations to find the cost of each apple and each banana. These are just a few examples of the many real-world applications of simultaneous equations. The ability to solve these systems is a valuable skill that can be applied in a wide range of contexts.
Conclusion
In conclusion, solving simultaneous equations is a crucial skill in mathematics with wide-ranging applications. In this article, we addressed the problem of solving the system 3x - 2y = 10 and x + 3y = 7, demonstrating two primary methods: substitution and elimination. The substitution method involves solving one equation for one variable and substituting that expression into the other equation, while the elimination method involves manipulating the equations to eliminate one variable through addition or subtraction. Both methods lead to the same solution, x = 4 and y = 1, showcasing their effectiveness. We provided step-by-step breakdowns of each method to ensure a clear understanding of the processes involved. Furthermore, we compared the two methods, highlighting their respective strengths and when each might be the preferred choice. Finally, we explored the real-world applications of simultaneous equations, illustrating their relevance in various fields such as physics, chemistry, economics, engineering, and even everyday scenarios. Mastering the techniques for solving simultaneous equations not only enhances mathematical proficiency but also equips individuals with valuable problem-solving skills applicable in diverse contexts. Whether you choose substitution, elimination, or a combination of both, the key is to practice and apply these methods to a variety of problems to solidify your understanding and build confidence. The ability to solve simultaneous equations is a fundamental building block for more advanced mathematical concepts and practical applications, making it an essential skill for students and professionals alike.
