Solving Systems Of Equations A Comprehensive Guide To 1.5x - 2.2y = -18 And 2.4x - 0.6y = 33
Hey everyone! Today, let's dive deep into the fascinating world of systems of equations. Specifically, we're going to tackle a problem that involves solving a system of two linear equations with two variables. This is a fundamental topic in mathematics, and mastering it will open doors to more advanced concepts in algebra and beyond. So, grab your pencils, and let's get started!
1. Understanding Systems of Equations
Before we jump into the nitty-gritty, let's make sure we're all on the same page about what a system of equations actually is. Simply put, a system of equations is a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all the equations in the system simultaneously. In our case, we're dealing with a system of two linear equations, which means each equation represents a straight line when graphed. The solution to the system is the point where these lines intersect.
The system we're working with today is:
1. 5x - 2.2y = -18
2. 4x - 0.6y = 33
Our mission, should we choose to accept it (and we do!), is to find the values of x and y that make both of these equations true. There are several methods we can use to solve systems of equations, including substitution, elimination, and graphing. We'll focus on the elimination method here, as it's particularly well-suited to this problem.
1.1 Why Systems of Equations Matter
You might be wondering, why bother with systems of equations at all? Well, they're incredibly useful for modeling real-world situations. Think about scenarios where you have multiple pieces of information and need to find unknown quantities. For example, you might have two different investments with different interest rates and want to figure out how much you invested in each. Or, you might be mixing two different solutions with different concentrations to achieve a desired concentration. Systems of equations provide a powerful tool for solving these kinds of problems.
1.2 Linear Equations Explained
Let's break down what makes an equation linear. A linear equation is an equation in which the highest power of any variable is 1. You'll notice in our system that both x and y appear without any exponents (or, more accurately, with an exponent of 1). This is what makes them linear. When you graph a linear equation, you always get a straight line. This geometric interpretation is crucial for understanding the solutions of systems of linear equations.
2. The Elimination Method A Step-by-Step Approach
The elimination method, also known as the addition method, is a technique for solving systems of equations by strategically adding or subtracting the equations to eliminate one of the variables. This leaves us with a single equation in one variable, which we can easily solve. Then, we can substitute that value back into one of the original equations to find the value of the other variable.
Here's how it works for our system:
2.1 Preparing the Equations
The first step in the elimination method is to make the coefficients of one of the variables opposites. Looking at our equations:
1. 5x - 2.2y = -18
2. 4x - 0.6y = 33
We can choose to eliminate either x or y. Let's choose to eliminate y. To do this, we need to find a common multiple of 2.2 and 0.6. The least common multiple of 22 and 6 (ignoring the decimal for a moment) is 66. So, we want to manipulate the equations so that the coefficients of y are 6.6 and -6.6 (or vice versa).
To achieve this, we can multiply the first equation by 0.3 and the second equation by -1.1. This gives us:
Equation 1 multiplied by 0.3: (1.5x - 2.2y = -18) * 0.3 -> 0.45x - 0.66y = -5.4
Equation 2 multiplied by -1.1: (2.4x - 0.6y = 33) * -1.1 -> -2.64x + 0.66y = -36.3
Notice how the coefficients of y are now opposites (-0.66 and +0.66). This is exactly what we wanted!
2.2 Eliminating the Variable
Now comes the fun part: adding the modified equations together. When we add the equations, the y terms will cancel out, leaving us with an equation in just x.
(0.45x - 0.66y = -5.4) + (-2.64x + 0.66y = -36.3) -> (0.45x - 2.64x) + (-0.66y + 0.66y) = -5.4 - 36.3
Simplifying, we get:
-2.19x = -41.7
2.3 Solving for x
Now we have a simple equation to solve for x. We just need to divide both sides by -2.19:
x = -41.7 / -2.19
x = 19
Great! We've found the value of x: it's 19.
2.4 Solving for y
Next, we need to find the value of y. To do this, we can substitute the value of x (19) back into either of the original equations. Let's use the first equation:
1. 5x - 2.2y = -18
Substituting x = 19, we get:
1. 5(19) - 2.2y = -18
2. 5 - 2.2y = -18
Now we solve for y:
-2.2y = -18 - 28.5
-2.2y = -46.5
y = -46.5 / -2.2
y = 21.14 (approximately)
So, the value of y is approximately 21.14.
3. Verifying the Solution The Importance of Checking Your Work
It's always a good idea to check your solution to make sure it's correct. To do this, we substitute the values of x and y back into both of the original equations and see if they hold true.
Let's start with the first equation:
1. 5x - 2.2y = -18
Substituting x = 19 and y = 21.14, we get:
1. 5(19) - 2.2(21.14) = -18
2. 5 - 46.508 = -18
-18.008 ≈ -18
That's pretty close! The slight difference is due to rounding the value of y. Now let's check the second equation:
2. 4x - 0.6y = 33
Substituting x = 19 and y = 21.14, we get:
3. 4(19) - 0.6(21.14) = 33
4. 6 - 12.684 = 33
33.016 ≈ 33
Again, the result is very close to 33, confirming that our solution is correct.
4. The Solution and Its Meaning
We've successfully solved the system of equations! Our solution is:
x = 19
y ≈ 21.14
This means that the point (19, 21.14) is the intersection point of the two lines represented by our equations. It's the only point that satisfies both equations simultaneously.
4.1 Visualizing the Solution
If we were to graph these two equations, we would see two lines intersecting at the point (19, approximately 21.14). This visual representation helps to solidify the concept of a solution to a system of equations as the point of intersection.
5. Alternative Methods A Quick Overview
While we focused on the elimination method, it's worth mentioning that there are other methods for solving systems of equations. Let's take a quick look at two of them:
5.1 Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This creates a single equation in one variable, which can be solved. Then, the value of that variable is substituted back into one of the original equations to find the other variable.
5.2 Graphing Method
The graphing method involves graphing both equations on the same coordinate plane. The solution to the system is the point where the lines intersect. This method is particularly useful for visualizing the solutions, but it may not be as accurate as the algebraic methods (elimination and substitution) if the solutions are not integers.
6. Conclusion Mastering the Art of Solving Systems
Solving systems of equations is a crucial skill in mathematics, with applications in various fields. We've tackled a specific example using the elimination method, but the concepts and techniques we've discussed apply more broadly. Remember the key steps:
- Prepare the equations by making the coefficients of one variable opposites.
- Eliminate the variable by adding the equations.
- Solve for the remaining variable.
- Substitute the value back into one of the original equations to find the other variable.
- Verify your solution.
By mastering these steps, you'll be well-equipped to solve a wide range of systems of equations. Keep practicing, and you'll become a system-solving pro in no time!
So, there you have it, guys! We've successfully navigated the world of systems of equations. Keep practicing, and you'll be solving these like a math whiz in no time. Until next time, happy problem-solving!