Solving Systems Of Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of systems of equations. Don't worry, it's not as scary as it sounds! We'll break down a specific problem step-by-step, so you can conquer these types of problems with confidence. Let's get started!

Understanding Systems of Equations

So, what exactly is a system of equations? In simple terms, it's a set of two or more equations that share the same variables. Our goal is to find the values of those variables that make all the equations true simultaneously. Think of it like finding the perfect key that unlocks all the locks in a set. In this particular case, we are presented with a system of two linear equations. These equations, when graphed, represent straight lines. The solution to the system is the point where these lines intersect, representing the (x, y) coordinates that satisfy both equations. Solving systems of equations is a fundamental skill in algebra and has wide-ranging applications in various fields, including physics, engineering, economics, and computer science. Mastering these techniques allows us to model real-world scenarios, make predictions, and solve complex problems involving multiple variables and constraints. From determining the optimal pricing strategy for a product to calculating the trajectory of a projectile, the ability to solve systems of equations opens doors to a multitude of analytical and problem-solving opportunities.

Our Problem: A Closer Look

Here's the system we'll be tackling:

-8y + 9x = -5
8y + 7x = -75
x = -5
y = ?

Notice that we already know the value of x! This makes our job much easier. We need to find the value of y that satisfies both equations, given that x = -5. The beauty of systems of equations lies in the fact that we have multiple equations relating the same variables. This redundancy allows us to use various techniques, such as substitution or elimination, to isolate and solve for the unknowns. In this case, knowing the value of x simplifies the process significantly. We can directly substitute this value into one or both equations to create a simpler equation with only y as the unknown. This approach highlights a crucial aspect of problem-solving: recognizing opportunities to simplify the problem by leveraging the information provided. By strategically using the given value of x, we can efficiently navigate the system of equations and determine the corresponding value of y. This not only showcases the power of substitution but also underscores the importance of careful observation and analysis in mathematical problem-solving.

Step-by-Step Solution

1. Substitution is Key

Since we know x = -5, let's substitute this value into the first equation:

-8y + 9(-5) = -5

Substitution is a powerful technique for solving systems of equations. It involves replacing one variable with its equivalent expression, effectively reducing the number of variables in an equation. In this instance, we have leveraged the known value of x to eliminate it from the equation, leaving us with an equation that solely involves y. This simplification is a cornerstone of many mathematical problem-solving strategies. By strategically substituting known values or expressions, we can transform complex equations into more manageable forms. The act of substitution allows us to focus on isolating and solving for a single variable, paving the way for a solution. This technique is not only applicable to systems of linear equations but also extends to more advanced mathematical concepts and real-world applications, making it a fundamental tool in any problem-solver's arsenal. The elegance of substitution lies in its ability to streamline the process, making seemingly daunting problems more accessible and solvable.

2. Simplify the Equation

Now, let's simplify:

-8y - 45 = -5

Simplification is an essential step in any mathematical problem-solving process. It involves applying algebraic rules and operations to reduce an equation or expression to its simplest form. In this case, we have simplified the equation by performing the multiplication operation and combining constant terms. This process not only makes the equation easier to read and understand but also sets the stage for isolating the variable we are trying to solve for. Simplification can involve a variety of techniques, including distributing terms, combining like terms, factoring, and reducing fractions. The goal is to eliminate unnecessary complexity and reveal the underlying structure of the equation. By simplifying, we can often gain a clearer understanding of the relationships between variables and constants, making it easier to identify the next steps in the solution process. This skill is not just confined to mathematics; it is a valuable asset in any field that requires logical thinking and problem-solving. The ability to break down complex information into manageable components is crucial for effective decision-making and critical analysis.

3. Isolate the 'y' Term

Add 45 to both sides:

-8y = 40

Isolating the variable is a critical step in solving equations. It involves manipulating the equation to get the variable term by itself on one side of the equation. In this case, we isolated the y term by adding 45 to both sides of the equation. This operation maintains the equality while effectively moving the constant term to the other side. The process of isolating a variable often involves a series of inverse operations, such as adding or subtracting, multiplying or dividing, to undo the operations that are currently affecting the variable. The goal is to create a situation where the variable term is the only term on one side of the equation, making it easier to determine its value. This technique is fundamental to solving a wide range of equations, from simple linear equations to more complex algebraic expressions. The ability to isolate variables is a testament to the power of algebraic manipulation and is an essential skill for anyone working with mathematical models and equations.

4. Solve for 'y'

Divide both sides by -8:

y = -5

Solving for the variable is the ultimate goal in equation solving. It involves performing the necessary operations to determine the value of the variable that satisfies the equation. In this instance, we solved for y by dividing both sides of the equation by -8. This operation isolates y and reveals its value, which is -5. The process of solving for a variable often requires a combination of algebraic techniques, including simplification, isolating terms, and applying inverse operations. It is a culmination of the steps taken to manipulate the equation into a form where the variable's value can be directly determined. The ability to accurately solve for variables is a fundamental skill in mathematics and has wide-ranging applications in science, engineering, economics, and other fields. It allows us to model real-world phenomena, make predictions, and solve problems involving unknown quantities. The satisfaction of finding the solution to an equation underscores the power of mathematical reasoning and the importance of perseverance in problem-solving.

The Answer!

So, we found that y = -5.

Therefore, the solution to the system of equations is:

x = -5
y = -5

Verification: The Key to Accuracy

To be absolutely sure of our answer, let's plug both x = -5 and y = -5 back into the second original equation (we already used the first one):

8y + 7x = -75
8(-5) + 7(-5) = -75
-40 - 35 = -75
-75 = -75

It checks out! This step is crucial to ensure that our solution is correct and satisfies all the equations in the system. Verification is a cornerstone of mathematical rigor and problem-solving. It involves substituting the obtained solution back into the original equations or conditions to confirm that it holds true. This process acts as a safeguard against errors that may have occurred during the solution process. By verifying our solution, we gain confidence in its accuracy and validity. In this case, we substituted the values of x and y back into the second original equation to ensure that it remains balanced. The fact that the equation holds true confirms that our solution is indeed correct. This practice of verification is not just limited to mathematical problems; it is a valuable habit in any field where accuracy and reliability are paramount. Whether it's checking the results of a scientific experiment, auditing financial statements, or debugging software code, verification plays a crucial role in ensuring the integrity of the final outcome.

Wrapping Up

Guys, solving systems of equations might seem tricky at first, but with a step-by-step approach and a little practice, you'll be solving them like a pro in no time! Remember the key techniques: substitution and simplification. And always, always verify your answer! Keep practicing, and you'll ace those math problems.