Solving Systems Of Equations X-3y=9 And Y=-x+5 A Step-by-Step Guide
Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of letters and numbers? Don't worry, we've all been there! Today, we're going to tackle a classic algebra problem: solving a system of equations. Specifically, we'll be diving deep into the equations X - 3y = 9 and y = -x + 5. Sounds intimidating? Trust me, it's not as scary as it looks. We'll break it down step-by-step, so you'll be solving these like a pro in no time!
Understanding the Problem: What are Systems of Equations?
So, what exactly are systems of equations? In simple terms, it's when you have two or more equations with the same variables (in our case, X and y). The goal is to find the values for those variables that make all the equations true at the same time. Think of it like a puzzle where you need to find the perfect pieces that fit together. In our example, we have two equations:
- X - 3y = 9
- y = -x + 5
Our mission, should we choose to accept it (and we do!), is to find the values of X and y that satisfy both of these equations simultaneously. There are several methods to solve these types of problems, but we will focus on the substitution method in this article. The substitution method is an effective way to solve systems of equations, particularly when one equation is already solved for one variable in terms of the other. It involves substituting the expression for one variable from one equation into the other equation. This process results in a single equation with one variable, which can then be solved. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is highly versatile and applicable to a wide range of linear and nonlinear systems of equations, making it a fundamental technique in algebra.
Before we jump into solving, it's crucial to grasp why these systems matter. They're not just abstract math problems; they pop up in real-world scenarios all the time! Imagine you're trying to figure out the cost of two different items given some combined prices, or maybe you're calculating the speed and distance of two moving objects. Systems of equations are the tools that help us model and solve these situations.
Key Takeaway: A system of equations is a set of two or more equations with the same variables, and our goal is to find the values for those variables that work for all equations.
The Substitution Method: Our Secret Weapon
Alright, now that we understand what we're dealing with, let's talk strategy. We're going to use the substitution method to crack this code. This method is super handy when one of your equations is already solved for one variable (like our second equation, y = -x + 5). The basic idea is to substitute the expression for one variable from one equation into the other equation. This might sound complicated, but trust me, it's easier than it sounds.
Think of it like this: if we know that y is equal to -x + 5, then we can replace every "y" in the first equation with "-x + 5". This will leave us with an equation that only has one variable (X), which we can then solve. So, let's get down to the nitty-gritty steps:
Step 1: Identify the Solved Variable
In our case, the second equation is already solved for y: y = -x + 5. This is perfect for the substitution method! Having one variable already isolated makes the process much smoother. It allows us to directly substitute the expression into the other equation, simplifying the problem significantly. Without this, we would have to rearrange one of the equations to isolate a variable, adding an extra step. This initial setup is a key advantage when using substitution.
Step 2: Substitute the Expression
Now, we'll substitute the expression for y (-x + 5) into the first equation (X - 3y = 9). This means we'll replace the "y" in the first equation with "(-x + 5)". Be careful to use parentheses to ensure you distribute correctly. This substitution is the heart of the method. By replacing one variable with its equivalent expression, we reduce the system to a single equation with a single variable. This transformation is what allows us to solve for one of the unknowns, paving the way to find the complete solution. It's a clever algebraic maneuver that simplifies a complex problem into a manageable one.
The equation will now look like this: X - 3(-x + 5) = 9
See? We've eliminated y from the first equation, and now we have an equation with only X. This is exactly what we wanted!
Key Takeaway: The substitution method involves replacing a variable in one equation with its equivalent expression from another equation. This simplifies the system and allows us to solve for one variable at a time.
Solving for X: Let's Get Algebraic!
Okay, we've set the stage, and now it's time to roll up our sleeves and do some algebra. Remember our equation after the substitution? It's X - 3(-x + 5) = 9. Our goal here is to isolate X on one side of the equation. To do that, we need to follow the order of operations and simplify step-by-step. This part is crucial, as any mistake here will affect the final answer. Accuracy and careful attention to detail are key to successfully solving for X. Each step builds upon the previous one, leading us closer to the solution.
Step 1: Distribute
First, we need to distribute the -3 across the parentheses. This means multiplying -3 by both -x and +5: X - 3(-x) - 3(5) = 9. Remember that multiplying two negatives results in a positive. This is a common area for errors, so it's important to pay close attention to the signs. The distribution step is essential for removing the parentheses and combining like terms, which is the next step in simplifying the equation. Without proper distribution, we cannot proceed further in solving for X.
This simplifies to: X + 3x - 15 = 9
Step 2: Combine Like Terms
Next, we'll combine the X terms on the left side of the equation: (1X + 3X) - 15 = 9. Combining like terms is a fundamental algebraic technique that simplifies equations by grouping similar terms together. This process reduces the number of terms in the equation, making it easier to manipulate and solve. In this case, combining the X terms allows us to isolate the variable more effectively in the subsequent steps.
This gives us: 4x - 15 = 9
Step 3: Isolate the Variable Term
Now, we want to get the term with X (4x) by itself on one side of the equation. To do this, we'll add 15 to both sides: 4x - 15 + 15 = 9 + 15. The addition property of equality states that adding the same value to both sides of an equation maintains the equality. This property is crucial for isolating variables and solving equations. By adding 15 to both sides, we eliminate the constant term on the left side, bringing us closer to isolating X.
This simplifies to: 4x = 24
Step 4: Solve for X
Finally, to get X by itself, we'll divide both sides of the equation by 4: (4x) / 4 = 24 / 4. The division property of equality, similar to the addition property, ensures that dividing both sides of an equation by the same non-zero value preserves the equality. This step is the final operation needed to isolate X and determine its value. Dividing by the coefficient of X completes the process of solving for this variable.
This gives us: X = 6
Woohoo! We've found the value of X. But we're not done yet ā we still need to find y.
Key Takeaway: Solving for X involves simplifying the equation by distributing, combining like terms, and isolating the variable using inverse operations (addition/subtraction and multiplication/division).
Finding Y: The Final Piece of the Puzzle
We've successfully solved for X, which is a major accomplishment! But remember, we're solving a system of equations, so we need to find the values for both X and y. Now that we know X = 6, finding y is a breeze. We can simply substitute the value of X back into either of our original equations. It's usually easiest to choose the equation that's already solved for y, which in our case is y = -x + 5. This equation is perfectly set up for us to plug in the value of X and calculate y directly. Using this equation minimizes the steps required and reduces the chance of errors.
Step 1: Substitute X into the Equation
We'll replace the "x" in the equation y = -x + 5 with the value we found, which is 6: y = -(6) + 5. Substitution is a powerful tool in algebra, allowing us to use known values to find unknowns. By substituting X = 6 into the equation, we transform it into a simple equation with only one variable, y. This step is the key to unlocking the value of y and completing the solution of the system of equations.
Step 2: Simplify and Solve for Y
Now, it's just a matter of simplifying: y = -6 + 5. This step involves basic arithmetic operations, addition in this case, to isolate and determine the value of y. Accurate calculation is crucial here to ensure we arrive at the correct solution for y. This final simplification brings us to the answer, completing the puzzle of solving the system of equations.
This gives us: y = -1
And there you have it! We've found that y = -1.
Key Takeaway: Once you've solved for one variable, substitute that value back into one of the original equations to solve for the other variable.
The Solution: X = 6, Y = -1
We did it! We've successfully solved the system of equations. Our solution is X = 6 and y = -1. But how do we know we're right? It's always a good idea to check your work to make sure your solution is correct. Verification is a crucial step in problem-solving, especially in mathematics. It ensures that the solution satisfies all the conditions of the problem. In the case of systems of equations, this means plugging the values of X and y back into the original equations to confirm that they hold true. This process helps to identify any potential errors and builds confidence in the accuracy of the solution.
Checking Our Solution
To check, we'll plug our values (X = 6 and y = -1) back into both of the original equations:
- X - 3y = 9 => 6 - 3(-1) = 9 => 6 + 3 = 9 => 9 = 9 (This checks out!)
- y = -x + 5 => -1 = -(6) + 5 => -1 = -6 + 5 => -1 = -1 (This checks out too!)
Since our values satisfy both equations, we can confidently say that our solution is correct.
Key Takeaway: The solution to a system of equations is a set of values for the variables that make all the equations true. Always check your solution by plugging the values back into the original equations.
Wrapping Up: You're a System Solver!
Awesome job, guys! You've just conquered a system of equations using the substitution method. Remember, the key is to break down the problem into smaller, manageable steps. Identify the solved variable, substitute the expression, solve for one variable, and then substitute again to find the other. And don't forget to check your work!
Systems of equations might seem tricky at first, but with practice, you'll become a whiz at solving them. These skills are valuable not only in math class but also in many real-world situations. So keep practicing, keep learning, and keep solving! You've got this!
