Solving X = 5y And X + Y = 60 System Of Equations A Step-by-Step Guide
Hey everyone! Today, we're diving into the exciting world of algebra to solve a system of equations. Don't worry, it's not as intimidating as it sounds. We'll break it down step by step so you can conquer these problems with confidence. Our mission? To find the values of x and y that satisfy both equations in the system:
- x = 5y
- x + y = 60
Understanding Systems of Equations
Before we jump into the solution, let's quickly recap what a system of equations is. Imagine you have two or more equations with two or more variables. The goal is to find the values for those variables that make all the equations true simultaneously. Think of it like a puzzle where all the pieces need to fit perfectly.
Systems of equations are a fundamental concept in mathematics, popping up in various real-world scenarios, from calculating mixtures in chemistry to modeling supply and demand in economics. Mastering these skills opens doors to solving a wide array of problems. In this specific case, we have two equations and two unknowns (x and y), which means we have a good chance of finding a unique solution. There are several methods to solve systems of equations, but we'll focus on the substitution method here because it's particularly well-suited for this problem.
Why the Substitution Method Works Wonders
The substitution method is a powerful technique because it allows us to simplify the problem by expressing one variable in terms of the other. In our case, the first equation (x = 5y) already gives us a direct relationship between x and y. This is a golden opportunity! We can substitute this expression for x into the second equation, effectively reducing the system to a single equation with a single variable. Once we solve for y, we can easily plug that value back into either of the original equations to find x. It's like a domino effect – one solution leads to the next!
This method is particularly useful when one of the equations is already solved for one variable, as we have here. It avoids the need for rearranging equations or performing more complex manipulations. However, even when an equation isn't explicitly solved for a variable, the substitution method can still be applied by first isolating one variable. This flexibility makes it a valuable tool in your problem-solving arsenal. The key is to identify the most straightforward path to simplify the system and solve for the unknowns.
Step-by-Step Solution
Okay, let's get down to business! Here's how we'll solve this system of equations using the substitution method:
Step 1: Substitute
Since we know that x = 5y, we can substitute 5y for x in the second equation:
(5y) + y = 60
Step 2: Simplify and Solve for y
Now we have a single equation with just one variable. Let's simplify and solve for y:
6y = 60
Divide both sides by 6:
y = 10
Step 3: Solve for x
Great! We've found the value of y. Now, let's plug it back into either of the original equations to find x. The first equation (x = 5y) looks easier, so let's use that:
x = 5 * 10
x = 50
Step 4: Verification (The Crucial Step!)
Before we declare victory, it's always a good idea to verify our solution. This is a critical step in solving any mathematical problem, and it helps prevent careless errors. We'll take our values for x and y ( x=50 and y=10) and plug them back into both original equations. If both equations hold true, then we know we've found the correct solution. If even one equation doesn't work, we need to retrace our steps and find the mistake. Verification is like the quality control checkpoint for our math work, ensuring accuracy and giving us confidence in our answer.
Let's start with the first equation: x = 5y. Substituting our values, we get 50 = 5 * 10, which simplifies to 50 = 50. This equation checks out! Now, let's move on to the second equation: x + y = 60. Substituting again, we get 50 + 10 = 60, which simplifies to 60 = 60. This equation also holds true. Since both equations are satisfied, we can confidently say that our solution is correct.
This verification process not only confirms our answer but also reinforces our understanding of the system of equations. It highlights the fact that the solution we found is not just a random set of numbers but a specific pair of values that simultaneously satisfy all the conditions laid out by the equations. By consistently verifying our solutions, we build good mathematical habits and reduce the chances of errors in future problems. So, always remember to double-check your work – it's the hallmark of a careful and successful problem-solver!
Step 5: State the Solution
Therefore, the solution to the system of equations is x = 50 and y = 10.
Alternative Methods for Solving Systems of Equations
While we've focused on the substitution method here, it's worth knowing that there are other techniques to tackle systems of equations. Two popular alternatives are the elimination method and the graphing method. Each method has its strengths and weaknesses, and the best approach often depends on the specific system you're dealing with.
Elimination Method: The Art of Canceling Variables
The elimination method, also known as the addition or subtraction method, is a powerful technique that aims to eliminate one of the variables by manipulating the equations. The core idea is to multiply one or both equations by suitable constants so that the coefficients of one of the variables become opposites (e.g., 2x and -2x). When you then add the equations together, that variable cancels out, leaving you with a single equation in one variable. This is particularly effective when the equations are in standard form (Ax + By = C), where the variables are aligned.
For instance, consider a system like:
2x + 3y = 7
4x - 3y = 5
Notice that the y coefficients are already opposites (3 and -3). By simply adding the two equations, the y terms will cancel out, and we'll be left with an equation in x. The elimination method can be a bit more involved if you need to multiply both equations to get the coefficients to match, but it's a valuable technique in your toolbox.
Graphing Method: Visualizing the Solution
The graphing method offers a visual approach to solving systems of equations. Each equation in the system represents a line when graphed on a coordinate plane. The solution to the system is the point where the lines intersect. This point represents the values of x and y that satisfy both equations simultaneously. To use this method, you typically need to rewrite each equation in slope-intercept form (y = mx + b), graph the lines, and then visually identify the intersection point.
The graphing method is excellent for understanding the nature of solutions. If the lines intersect at a single point, the system has a unique solution. If the lines are parallel, they never intersect, and the system has no solution. If the lines coincide (are the same line), there are infinitely many solutions. However, the graphing method may not be the most accurate for finding solutions with non-integer values, as reading the exact coordinates of the intersection point from a graph can be challenging. Still, it's a valuable tool for visualizing the solutions and understanding the relationships between the equations.
Real-World Applications of Systems of Equations
Systems of equations aren't just abstract mathematical concepts; they're powerful tools that help us model and solve real-world problems in various fields. From everyday scenarios like calculating costs and distances to complex scientific and engineering applications, systems of equations play a crucial role. Understanding how to set up and solve these systems allows us to make informed decisions and find optimal solutions.
Mixing Solutions and Allocating Resources
One common application involves mixing solutions with different concentrations. For example, imagine a chemist needs to create a specific amount of a solution with a certain concentration by mixing two solutions with different concentrations. Systems of equations can be used to determine the exact volumes of each solution needed to achieve the desired result. Similarly, businesses often use systems of equations to allocate resources, such as budget or manpower, across different projects or departments to maximize efficiency and profitability. These problems often involve setting up equations based on constraints and requirements, and then solving for the optimal allocation using techniques like substitution or elimination.
Calculating Distances, Rates, and Time
Another frequent application is in problems involving distances, rates, and time. For instance, consider a scenario where two cars are traveling towards each other from different locations. We can use a system of equations to determine when and where they will meet, given their speeds and initial distances. These types of problems often involve using the formula distance = rate × time to set up the equations. Systems of equations are also used in navigation and aviation to calculate flight paths, fuel consumption, and estimated arrival times. The ability to accurately model these scenarios is critical for safety and efficiency in transportation.
Supply and Demand Equilibrium in Economics
In economics, systems of equations are used to model the relationship between supply and demand. The demand equation represents the quantity of a product or service that consumers are willing to buy at a given price, while the supply equation represents the quantity that producers are willing to sell at that price. The equilibrium point, where the supply and demand curves intersect, represents the price and quantity at which the market is in balance. Systems of equations can be used to find this equilibrium point, which is crucial for understanding market dynamics and making pricing decisions. These models can also be extended to analyze the effects of government policies, such as taxes and subsidies, on market outcomes. The applications in economics are vast, ranging from microeconomic analysis of individual markets to macroeconomic models of national economies.
Conclusion
So, there you have it! We've successfully solved the system of equations x = 5y and x + y = 60 using the substitution method. Remember, the key is to break down the problem into manageable steps and verify your solution. With practice, you'll become a system-of-equations-solving pro! Keep practicing, and don't be afraid to explore different methods and applications. You've got this! Remember, math is like building with LEGOs – each piece (concept) fits together to create something amazing. Keep building your mathematical skills, one equation at a time, and you'll be surprised at what you can create!
