Solving Systems Of Equations: A Step-by-Step Guide
Introduction: Understanding Systems of Equations
In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering to economics. When you encounter a set of two or more equations with the same variables, you're dealing with a system of equations. The solution to such a system is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point where all the lines (or curves, in more complex systems) intersect on a graph. This guide will walk you through a straightforward method to solve a specific system of linear equations. Our main focus will be on understanding the substitution method, its applications, and why it's a powerful tool in algebra. Let's dive deep into the world of systems of equations and learn how to conquer them!
Solving systems of equations is crucial not only in academic settings but also in real-world applications. Understanding how to find the common solution helps in modeling various scenarios where multiple conditions must be met simultaneously. For instance, in economics, it can help determine the equilibrium price and quantity in a market. In engineering, it can be used to design circuits or structures where different constraints apply. The system we're addressing, y = -5x + 6 and y = 3x - 2, is a classic example of a linear system, where both equations represent straight lines. The point where these lines intersect is the solution we seek. By mastering techniques like substitution, you'll be equipped to tackle a wide array of mathematical problems.
Before we proceed, it's important to grasp the underlying principle: we're looking for the x and y values that make both equations true at the same time. Graphically, this corresponds to the intersection point of the two lines. Algebraically, we can achieve this by manipulating the equations to isolate one variable and then substituting its expression into the other equation. This process reduces the problem to a single equation with a single variable, which is much easier to solve. As we move through the steps, pay close attention to how each operation helps us get closer to the solution. Remember, math isn't just about memorizing steps; it's about understanding why those steps work. This understanding will make you a much more effective problem solver in the long run.
Step 1: Recognize the Equations
Our system of equations consists of two linear equations:
- y = -5x + 6
- y = 3x - 2
The beauty of this system is that both equations are already solved for y. This makes the substitution method particularly straightforward. When you encounter a system where equations are not explicitly solved for one variable, your first step might involve rearranging them to isolate a variable. However, in our case, we can jump right into the substitution process. Recognizing the structure of the equations is a crucial initial step, as it guides our choice of method and simplifies the subsequent steps.
When approaching any system of equations, it's essential to first identify the type of equations you're dealing with. Are they linear, quadratic, or something else? In our case, both equations are linear, meaning they represent straight lines when graphed. This simplifies the problem, as we know there will be at most one solution (one intersection point) unless the lines are parallel or coincident. Recognizing this linearity helps us anticipate the nature of the solution and choose an appropriate method. The fact that both equations are already solved for y suggests that substitution will be a highly efficient approach.
Furthermore, notice how the equations are structured in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. This form provides valuable insights into the behavior of the lines. For instance, we can see that the first line has a negative slope (-5), meaning it slopes downward from left to right, while the second line has a positive slope (3), meaning it slopes upward. This confirms that the lines are not parallel and will indeed intersect at a single point. By carefully observing the structure and form of the equations, we can often gain valuable clues that simplify the solution process.
Step 2: Apply the Substitution Method
Since both equations are solved for y, we can set them equal to each other:
-5x + 6 = 3x - 2
This step is the heart of the substitution method. We're essentially saying that at the point where the two lines intersect, the y values must be the same. Therefore, the expressions representing y in both equations must be equal. This transforms our two-equation system into a single equation with only one variable (x), which is much easier to solve. The key to successful substitution lies in accurately equating the expressions and then carefully manipulating the resulting equation.
The substitution method works because it leverages the fundamental principle that the solution to a system of equations must satisfy all equations simultaneously. By setting the two expressions for y equal to each other, we're enforcing this condition. We're essentially finding the x value for which the two lines have the same y value, which corresponds to their point of intersection. This approach is particularly powerful when one or both equations are already solved for a variable, as it minimizes the need for algebraic manipulation in the initial steps.
It's crucial to understand the logic behind this step rather than simply memorizing it. When you grasp the concept that we're equating the y values at the point of intersection, the substitution method becomes intuitive and less prone to errors. Also, note that this method can be applied to systems with more than two equations and variables, although the complexity increases. The core idea remains the same: eliminate variables by substituting expressions until you have a single equation with a single unknown.
Step 3: Solve for x
Now, let's solve the equation for x:
-5x + 6 = 3x - 2
Add 5x to both sides:
6 = 8x - 2
Add 2 to both sides:
8 = 8x
Divide both sides by 8:
x = 1
This is a crucial step where accuracy is paramount. Each algebraic manipulation must be performed carefully to avoid errors that can lead to an incorrect solution. The goal is to isolate x on one side of the equation, and we achieve this by systematically applying inverse operations. Adding 5x to both sides cancels out the -5x term on the left, and adding 2 to both sides cancels out the -2 on the right. Finally, dividing both sides by 8 isolates x. It's always a good practice to double-check each step to ensure no mistakes were made.
Solving for x is a critical milestone because it gives us one coordinate of the solution point. Remember, the solution to a system of two equations in two variables is an ordered pair (x, y) that satisfies both equations. We've now found the x-coordinate, which is 1. This means that the intersection point of the two lines has an x-coordinate of 1. The next step will be to find the corresponding y-coordinate. The value of x we've found is the only x value that will make the two expressions for y equal, ensuring that the solution we find will satisfy both equations simultaneously.
Before moving on, it's worthwhile to pause and reflect on the process. We started with two equations and two unknowns, and through substitution and algebraic manipulation, we've reduced the problem to a single equation with one unknown. This is a common strategy in problem-solving: break down a complex problem into simpler steps. By focusing on isolating x, we've made significant progress toward finding the complete solution. Now, we'll use this value of x to find the corresponding value of y.
Step 4: Substitute x Back into One of the Original Equations
We can substitute x = 1 into either equation to find y. Let's use the first equation:
y = -5x + 6
y = -5(1) + 6
y = -5 + 6
y = 1
This step is the final piece of the puzzle. Now that we have the x-coordinate of the solution, we can substitute it back into either of the original equations to find the corresponding y-coordinate. The choice of which equation to use is arbitrary; both will yield the same result if our calculations are correct. However, it's often wise to choose the equation that appears simpler or easier to work with. In this case, the first equation, y = -5x + 6, seems slightly simpler due to the smaller coefficients.
Substituting x back into the equation is a crucial step because it allows us to find the y-coordinate that corresponds to the x-coordinate we found earlier. This ensures that the ordered pair (x, y) satisfies both equations simultaneously. It's like finding the specific point on each line that has the same x value, which is the point of intersection. The arithmetic in this step is straightforward, but it's essential to perform it carefully to avoid errors. Substituting x = 1 into y = -5x + 6 yields y = -5(1) + 6 = -5 + 6 = 1. Therefore, the y-coordinate of the solution is 1.
We now have both coordinates of the solution: x = 1 and y = 1. This means that the point (1, 1) is the intersection point of the two lines represented by the original equations. This point satisfies both equations, meaning that if we substitute these values into either equation, we will get a true statement. Before we conclude, it's always a good idea to verify our solution by substituting both x and y values into both original equations to ensure accuracy.
Step 5: State the Solution
The solution to the system of equations is x = 1 and y = 1, or the ordered pair (1, 1).
This is the culmination of our efforts. We've systematically worked through the problem, applying the substitution method and algebraic manipulations to arrive at the solution. Stating the solution clearly is essential because it provides a definitive answer to the question. The solution to a system of two equations in two variables is an ordered pair (x, y) that represents the point of intersection of the two lines. In this case, the solution is (1, 1), which means that the two lines intersect at the point with coordinates x = 1 and y = 1.
It's helpful to think of this solution graphically. If we were to plot the two lines represented by the equations y = -5x + 6 and y = 3x - 2, we would see that they intersect at the point (1, 1). This visual representation can provide a deeper understanding of what the solution represents. It's not just a pair of numbers; it's a specific point in the coordinate plane where both equations hold true.
Before we consider our solution final, let's perform a crucial step: verification. We'll substitute the values x = 1 and y = 1 into both original equations to ensure they are satisfied. This will confirm that our solution is correct and that we haven't made any errors along the way. This step is highly recommended for every system of equations problem, as it can catch mistakes and give you confidence in your answer.
Step 6: Verify the Solution
Let's check our solution by substituting x = 1 and y = 1 into both original equations:
Equation 1: y = -5x + 6
1 = -5(1) + 6
1 = -5 + 6
1 = 1 (True)
Equation 2: y = 3x - 2
1 = 3(1) - 2
1 = 3 - 2
1 = 1 (True)
Since the solution satisfies both equations, it is correct.
Verifying the solution is an essential step in any mathematical problem, especially when solving systems of equations. It provides a concrete way to check the accuracy of our work and ensures that the solution we've found truly satisfies all the given conditions. In this case, we substitute the values x = 1 and y = 1 into both original equations and verify that the resulting statements are true. If even one equation is not satisfied, it indicates an error in our calculations, and we would need to go back and review our steps.
By substituting x = 1 and y = 1 into the first equation, y = -5x + 6, we get 1 = -5(1) + 6, which simplifies to 1 = -5 + 6, and finally, 1 = 1. This is a true statement, confirming that the solution satisfies the first equation. Similarly, substituting the values into the second equation, y = 3x - 2, we get 1 = 3(1) - 2, which simplifies to 1 = 3 - 2, and again, 1 = 1. This also confirms that the solution satisfies the second equation.
Since the solution (1, 1) satisfies both equations, we can confidently conclude that it is the correct solution to the system. This verification step not only gives us assurance in our answer but also reinforces the understanding of what it means for a solution to satisfy a system of equations. It's a valuable habit to develop in mathematics, as it can prevent errors and improve problem-solving skills.
Conclusion
We have successfully solved the system of equations using the substitution method. The solution is x = 1 and y = 1, or the ordered pair (1, 1). Remember, solving systems of equations is a fundamental skill with wide-ranging applications. By mastering techniques like substitution, you'll be well-equipped to tackle a variety of mathematical and real-world problems.
In conclusion, understanding the substitution method and its application is crucial for solving systems of equations. This method involves solving one equation for one variable and then substituting that expression into the other equation, thereby reducing the system to a single equation with a single variable. We've seen how this approach works step-by-step, from recognizing the equations to verifying the solution. The ability to solve systems of equations is a valuable skill that extends beyond the classroom and into various fields such as engineering, economics, and computer science.
Throughout this guide, we've emphasized the importance of each step and the logic behind it. Solving math problems isn't just about following a set of rules; it's about understanding the underlying principles and applying them effectively. By practicing these techniques and understanding the concepts, you'll develop confidence and proficiency in solving systems of equations. Remember to always verify your solutions to ensure accuracy and reinforce your understanding.
To further enhance your understanding of systems of equations and explore other methods of solving them, consider visiting reputable online resources. A great place to start is Khan Academy's Systems of Equations section, where you can find comprehensive lessons, practice exercises, and videos that cover various aspects of this topic.
