Solving Systems Of Equations A Step By Step Guide To 2x + 3y = 35 And X + Y = 13

Jul 30, 2025 by ADMIN 81 views

Unraveling the Mystery The Solution to the System of Equations 2x + 3y = 35 and x + y = 13

Hey there, math enthusiasts! Ever find yourself staring at a system of equations, feeling like you're trying to crack a secret code? Well, you're not alone! Systems of equations might seem intimidating at first, but once you understand the underlying concepts, they become a fun puzzle to solve. In this article, we're going to break down a classic system of equations problem step by step, so you can confidently tackle similar problems in the future. Our mission today is to find the solution to the system of equations: 2x + 3y = 35 and x + y = 13. We will explore different methods to solve this system, providing clear explanations and examples along the way. So grab your pencil and paper, and let's dive into the world of algebra!

Understanding Systems of Equations

Before we jump into solving our specific system, let's take a moment to understand what a system of equations actually is. Guys, think of it like this: a system of equations is simply a set of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. In our case, we have two equations with two variables, x and y. The solution to the system will be a pair of values (x, y) that makes both equations true. This means that when you substitute these values into both equations, the left-hand side will equal the right-hand side. This concept is crucial because it forms the basis for all the methods we'll use to solve the system. The beauty of a system of equations lies in its ability to model real-world situations. For example, we might use a system of equations to determine the cost of two different items given their combined price and the price difference. Or, we could use it to calculate the speeds of two cars traveling towards each other. The possibilities are endless! By mastering the techniques to solve these systems, you're not just learning math; you're gaining a powerful tool for problem-solving in various contexts. Always remember, the key is to find the values that work for every equation in the system, ensuring that all conditions are met. Understanding this foundational principle will make the process of solving systems of equations much more intuitive and less daunting.

Method 1: The Substitution Method

The substitution method is one of the most versatile techniques for solving systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This will leave us with a single equation with a single variable, which we can easily solve. Let's apply this to our system:

Choose an equation and solve for one variable: Looking at our equations, x + y = 13 seems simpler to work with. Let's solve this equation for x: x = 13 - y This step is crucial because it allows us to express one variable in terms of the other. By isolating x, we've created a new expression that we can plug into the other equation. This is the heart of the substitution method – turning a two-variable problem into a one-variable problem. When choosing which equation and which variable to solve for, always look for the path of least resistance. Equations with coefficients of 1 are generally easier to manipulate. In this case, solving x + y = 13 for x was a straightforward process. This strategic choice simplifies the subsequent steps and reduces the chances of making errors. So, always take a moment to assess your options and pick the most efficient route.
Substitute: Now we'll substitute this expression for x into the other equation, 2x + 3y = 35: 2(13 - y) + 3y = 35 Substituting the expression we found for x into the second equation is a pivotal step. This is where we eliminate one variable and consolidate the problem into a single equation. By replacing x with (13 - y), we've transformed the equation into one that only involves y. This allows us to focus on solving for y first, and then we can easily find x later. It's like solving a puzzle – you take the pieces you know and use them to reveal the missing ones. The substitution method is powerful because it systematically reduces the complexity of the system, making it solvable. Always double-check your substitution to ensure accuracy. A small error in this step can throw off the entire solution. Precision and attention to detail are key to success with this method.
Solve for y: Now we have a simple equation with just one variable. Let's solve for y:
```
26 - 2y + 3y = 35
y = 9
```
Solving for y in the equation we obtained after substitution is a straightforward algebraic process. We start by distributing, then combine like terms, and finally isolate y on one side of the equation. Each step is a building block towards finding the value of y. The beauty of this stage is that it brings us closer to the solution with each manipulation. Once we have the value of y, we can use it to find the value of x. This highlights the interconnectedness of the variables in a system of equations. The solution for one variable directly impacts the solution for the other. So, each time you solve for a variable, you're not just finding a number; you're uncovering a crucial piece of the puzzle. Always remember to perform these algebraic manipulations carefully, paying attention to signs and operations. A methodical approach will ensure accuracy and prevent common errors. With the value of y in hand, we're ready to move on to the final step and complete the solution.
Solve for x: Now that we know y = 9, we can substitute this value back into either of the original equations or the equation x = 13 - y. Let's use the latter: x = 13 - 9 x = 4 After finding the value of y, substituting it back into one of the original equations or the derived equation (like x = 13 - y) is the final step in solving for x. This is where we bring everything together and complete the solution pair (x, y). Choosing the easiest equation for substitution can save time and reduce the chances of errors. In this case, using x = 13 - y was the most direct route. The beauty of this step is that it solidifies our understanding of the system of equations. We're not just finding individual values; we're finding a pair of values that satisfy both equations simultaneously. This is the essence of solving a system – finding the common ground between the equations. Always remember to double-check your solution by substituting both x and y values into both original equations. This is the ultimate test of accuracy and confirms that you've truly solved the system. With both x and y determined, we have successfully unlocked the solution to the puzzle!
Solution: Therefore, the solution to the system of equations is x = 4 and y = 9. This can be written as the ordered pair (4, 9). The final step in solving a system of equations is to clearly state the solution. This is crucial because it communicates your findings in a concise and unambiguous way. The solution is typically expressed as an ordered pair (x, y), which represents the values that satisfy both equations in the system. Writing the solution in this format helps to emphasize that we've found a pair of values that work together. It's not just about finding x and y individually; it's about finding the combination that makes the entire system true. Presenting the solution as an ordered pair also makes it easy to visualize the solution graphically. Each equation in the system represents a line, and the solution (x, y) is the point where these lines intersect. This visual representation can provide a deeper understanding of what it means to solve a system of equations. Always remember to double-check your solution by substituting the values back into the original equations. This is the ultimate verification that you've found the correct answer. With the solution clearly stated and verified, you can confidently move on to the next challenge!

Method 2: The Elimination Method

Another powerful method for solving systems of equations is the elimination method. In this technique, we manipulate the equations so that when we add them together, one of the variables is eliminated. This simplifies the system, allowing us to solve for the remaining variable. Let's see how this works with our equations:

Multiply equations (if necessary): Our goal is to have the coefficients of either x or y be opposites. Let's eliminate x. We can multiply the second equation (x + y = 13) by -2: -2(x + y) = -2(13) -2x - 2y = -26 The first critical step in the elimination method is to manipulate the equations so that the coefficients of one variable are opposites. This is essential because when we add the equations together, that variable will be eliminated, simplifying the system. In our case, we chose to eliminate x. By multiplying the second equation by -2, we created a coefficient of -2 for x, which is the opposite of the coefficient of x in the first equation (which is 2). This strategic move sets the stage for the elimination to occur. When deciding which variable to eliminate, look for the easiest path. Sometimes, multiplying just one equation is sufficient, while other times, you may need to multiply both equations. The key is to create opposite coefficients for the variable you want to eliminate. Always remember to distribute the multiplication to every term in the equation. A small error in this step can derail the entire process. With the equations properly manipulated, we're now ready to add them together and eliminate x.
Add the equations: Now we can add the modified second equation to the first equation:
```
2x + 3y = 35
-2x - 2y = -26
----------------
y = 9
```
Adding the equations together is the heart of the elimination method. This is where the magic happens and one of the variables disappears. By carefully manipulating the equations in the previous step, we ensured that the coefficients of x were opposites. When we add the equations vertically, the x terms cancel out, leaving us with an equation in terms of y only. This simplification is the key to the elimination method's effectiveness. It transforms a two-variable problem into a single-variable problem, which is much easier to solve. The process of adding the equations involves combining like terms vertically. Make sure to add the x terms together, the y terms together, and the constant terms together. Pay close attention to signs – a simple mistake can lead to an incorrect result. After adding the equations, we are left with y = 9. This is a significant step forward, as we have now found the value of one of the variables. With the value of y in hand, we can proceed to the next step and solve for x.
Solve for x: Now that we know y = 9, we can substitute this value back into either of the original equations. Let's use x + y = 13: x + 9 = 13 x = 4 After successfully eliminating one variable and solving for the other, the next step is to substitute the known value back into one of the original equations to find the remaining variable. This is where we leverage the information we've gained to complete the solution. Choosing the simplest equation for substitution can make the process easier and reduce the risk of errors. In this case, x + y = 13 was a straightforward choice. By substituting y = 9 into this equation, we obtain a simple equation in terms of x, which we can easily solve. This substitution step highlights the interconnectedness of the variables in a system of equations. Once we know the value of one variable, we can use that information to find the value of the other. It's like a domino effect – each piece of information leads us closer to the complete solution. Always double-check your substitution to ensure accuracy. A small mistake in this step can lead to an incorrect final answer. With both x and y determined, we have successfully solved the system of equations using the elimination method.
Solution: Therefore, the solution to the system of equations is x = 4 and y = 9, or (4, 9). The final step in solving a system of equations, regardless of the method used, is to clearly state the solution. This is crucial for communicating your findings effectively and ensuring that the answer is easily understood. The solution is typically presented as an ordered pair (x, y), where x and y represent the values that satisfy both equations in the system. This notation emphasizes that we have found a pair of values that work together, not just individual solutions for each variable. Writing the solution as an ordered pair also aligns with the graphical interpretation of systems of equations. Each equation represents a line, and the solution is the point where the lines intersect. This visual representation can help to solidify your understanding of what it means to solve a system. Always remember to double-check your solution by substituting the values of x and y back into the original equations. This is the ultimate verification that you have found the correct answer. With the solution clearly stated and verified, you can confidently move on to the next mathematical challenge! The ability to clearly and accurately present your solutions is a vital skill in mathematics and beyond.

Method 3: Graphical Method

Visual learners, this one's for you! The graphical method provides a visual way to solve systems of equations. Each equation represents a line on a graph, and the solution to the system is the point where the lines intersect. Let's graph our equations:

Rewrite the equations in slope-intercept form (y = mx + b):
- 2x + 3y = 35 becomes y = (-2/3)x + 35/3
- x + y = 13 becomes y = -x + 13 Transforming equations into slope-intercept form (y = mx + b) is a crucial step in the graphical method. This form makes it easy to visualize and plot the lines represented by the equations. The slope (m) tells us the steepness and direction of the line, while the y-intercept (b) tells us where the line crosses the y-axis. By rewriting our equations in this form, we gain valuable insights into their graphical behavior. We can quickly identify the slope and y-intercept of each line, which are essential for plotting them on a graph. The process of converting to slope-intercept form involves isolating y on one side of the equation. This requires careful algebraic manipulation, including adding or subtracting terms and dividing by coefficients. Always double-check your work to ensure that you have correctly transformed the equations. A mistake in this step can lead to an incorrect graph and, consequently, an incorrect solution. With the equations in slope-intercept form, we're ready to move on to the next step and plot the lines on a coordinate plane.
Graph the lines: Plot the two lines on the same coordinate plane. You can use the slope and y-intercept to help you graph each line, or you can find two points on each line and connect them. Graphing the lines is the visual centerpiece of the graphical method. Each equation in the system represents a line on the coordinate plane, and the point where these lines intersect is the solution to the system. Plotting the lines accurately is crucial for finding the correct solution. There are several ways to graph a line. One common method is to use the slope-intercept form (y = mx + b). The y-intercept (b) gives you one point on the line, and the slope (m) tells you how to find other points. Another method is to find two points that satisfy the equation and connect them. For example, you can choose two values for x, plug them into the equation, and solve for y. This will give you two points (x, y) that you can plot. When graphing the lines, use a ruler or straightedge to ensure accuracy. A slight deviation in the line can lead to an incorrect intersection point and, therefore, an incorrect solution. Pay close attention to the scales on the axes and choose appropriate increments to make the graph clear and readable. With the lines accurately plotted, we can visually identify the point of intersection, which represents the solution to the system.
Find the intersection point: The point where the two lines intersect is the solution to the system of equations. In this case, the lines intersect at the point (4, 9). Identifying the intersection point is the key to solving a system of equations graphically. The point where the lines intersect represents the solution because it is the only point that lies on both lines simultaneously. This means that the coordinates of the intersection point satisfy both equations in the system. To find the intersection point, carefully observe the graph and estimate the coordinates of the point where the lines cross. In some cases, the intersection point will be a clear and obvious point on the grid. In other cases, you may need to estimate the coordinates based on the lines' positions. For more precise solutions, you can use graphing software or calculators that can find the intersection point automatically. However, it's essential to understand the underlying concept – the intersection point represents the solution. Once you've identified the intersection point, double-check your solution by substituting the coordinates into the original equations. This will verify that the point truly satisfies both equations. The graphical method provides a visual and intuitive way to understand systems of equations. It allows you to see the solution as a point in a coordinate plane, making the concept more concrete.
Solution: Thus, the solution to the system of equations is x = 4 and y = 9. The final step in the graphical method, just as with the other methods, is to clearly state the solution. This ensures that your findings are communicated effectively and are easy to understand. The solution is typically presented as the values of x and y that correspond to the intersection point of the lines. It's crucial to connect the visual representation (the intersection point) with the algebraic solution (the values of x and y). This reinforces the understanding that the solution is the point that satisfies both equations in the system. In this case, the intersection point is (4, 9), so the solution is x = 4 and y = 9. Always remember to double-check your solution by substituting these values back into the original equations. This is the ultimate verification that you have found the correct answer. The graphical method provides a powerful visual tool for solving systems of equations. It allows you to see the relationship between the equations and their solutions in a geometric context. By understanding the graphical method, you gain a deeper appreciation for the concepts underlying systems of equations.

Verifying the Solution

No matter which method we use, it's crucial to verify our solution. To do this, we substitute x = 4 and y = 9 back into the original equations:

2x + 3y = 35 2(4) + 3(9) = 8 + 27 = 35 (Correct!)
x + y = 13 4 + 9 = 13 (Correct!)

Since the values satisfy both equations, we can confidently say that our solution is correct.

Verifying the solution is an indispensable step in solving systems of equations (and in mathematics in general). It's the ultimate check that ensures our hard work has paid off and that we've found the correct answer. The process of verification involves substituting the values we've obtained for the variables back into the original equations. If the values satisfy all the equations in the system, then we can confidently declare that our solution is correct. This step is not just about getting the right answer; it's also about building confidence in our problem-solving abilities. By verifying our solution, we reinforce our understanding of the concepts and techniques involved. It also helps to catch any errors we may have made along the way. Sometimes, a small mistake in the calculations can lead to an incorrect solution. Verification helps us to identify and correct these errors before we move on. In addition to numerical verification, it's also helpful to think about the reasonableness of the solution in the context of the problem. Do the values make sense? Are they consistent with the information given? This kind of logical reasoning can provide an additional layer of assurance. So, always make verification a habit. It's a simple step that can save you from making costly mistakes and help you to become a more confident and proficient problem solver. With our solution verified, we can move forward knowing that we've successfully tackled the problem.

Conclusion

So there you have it! We've successfully solved the system of equations 2x + 3y = 35 and x + y = 13 using three different methods: substitution, elimination, and graphing. We found that the solution is x = 4 and y = 9, or the ordered pair (4, 9). Remember, guys, practice makes perfect! The more you work with systems of equations, the more comfortable and confident you'll become. Feel free to try these methods on other systems of equations. And don't be afraid to explore other mathematical concepts and problems. The world of mathematics is vast and fascinating, and there's always something new to learn!

In conclusion, mastering the solution of systems of equations is a valuable skill in mathematics and beyond. It's not just about finding the right numbers; it's about developing problem-solving strategies, logical reasoning, and analytical thinking. The three methods we've explored – substitution, elimination, and graphing – offer different approaches to tackling these problems. Each method has its strengths and weaknesses, and the best method for a particular problem may depend on the specific equations involved. By understanding all three methods, you'll be well-equipped to solve a wide range of systems of equations. Remember, the key to success is practice. The more you work through examples and apply these methods, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're a natural part of the learning process. And don't hesitate to seek help from teachers, classmates, or online resources when you encounter difficulties. The journey of learning mathematics is a rewarding one, and the skills you develop along the way will serve you well in many areas of life. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics awaits!