Solving Systems Of Equations: A Step-by-Step Guide
Hey guys! Ever stumbled upon a system of equations and felt a bit lost? Don't worry, you're not alone! Systems of equations might seem intimidating at first, but with the right approach, they can be cracked. In this guide, we'll break down a specific system of equations step by step, making it super easy to understand. We'll focus on the system:
y = 2x - 3.5
x - 2y = -14
So, let's dive in and learn how to solve this like a pro!
Understanding Systems of Equations
Before we jump into solving, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations that share the same variables. Our goal is to find the values for those variables that make all the equations in the system true simultaneously. Think of it like finding the sweet spot where all the equations agree.
In our case, we have two equations with two variables, x and y. This is a pretty common scenario, and there are several methods we can use to find the solution. We'll be focusing on the substitution method here, as it's particularly effective for this type of system. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable and leaves us with a single equation that we can solve. This first key step in dealing with systems of equations ensures a solid foundation for finding accurate solutions. Understanding the interrelation between equations allows for strategic manipulation and simplification, ultimately revealing the values of the variables involved. So, before diving into calculations, let's make sure we understand the basics. Grasping these concepts will not only simplify the solving process but also enrich our problem-solving capabilities across various mathematical landscapes. Remember, each equation is a piece of the puzzle; understanding their interplay is what leads us to the final picture.
Step 1: Choose an Equation and Solve for a Variable
The first step in the substitution method is to pick one of the equations and solve it for one of the variables. Looking at our system:
y = 2x - 3.5
x - 2y = -14
The first equation, y = 2x - 3.5, looks like a great starting point because y is already isolated on one side. This means we don't have to do any extra work to get a variable by itself – it's already done for us! This is a crucial observation, as it significantly streamlines the solving process. Recognizing such opportunities to simplify the initial steps is a hallmark of effective problem-solving in mathematics. By selecting the equation where a variable is already isolated, we can avoid unnecessary algebraic manipulations and immediately proceed to the next phase: substitution. Remember, identifying these pre-existing conditions not only saves time but also reduces the chances of making errors. So, let’s leverage this advantage and seamlessly transition to the next stage, where we’ll utilize this equation to unlock the solution to our system.
Step 2: Substitute the Expression into the Other Equation
Now comes the substitution part – the heart of this method! We know that y is equal to 2x - 3.5. So, we can take this expression and substitute it for y in the other equation, which is x - 2y = -14. This is where the magic happens! By replacing y with its equivalent expression, we effectively eliminate y from the second equation, leaving us with an equation that only involves x. This transformation is pivotal because it allows us to solve for x directly. The substitution method cleverly leverages the relationship between the equations to simplify the problem. By replacing one variable with an expression involving the other, we reduce the complexity and make the equation solvable. This technique is a cornerstone of algebraic problem-solving and demonstrates the power of strategic manipulation. Let's perform the substitution carefully and witness how it unlocks the path to our solution.
So, let's do it. Replacing y in the second equation, we get:
x - 2(2x - 3.5) = -14
Step 3: Solve for the Remaining Variable
We've now got an equation with only one variable, x. Our next step is to solve for x. This involves simplifying the equation and isolating x on one side. Remember those algebra skills? They're about to come in handy! We'll need to distribute, combine like terms, and use inverse operations to get x all by itself. This step is crucial because it's where we actually find the value of one of our variables. Mastering this process ensures accurate solutions and builds confidence in tackling more complex equations. Let’s roll up our sleeves and work through the algebra to uncover the value of x. This is where the puzzle pieces start to fall into place, bringing us closer to the complete solution of our system.
First, we distribute the -2:
x - 4x + 7 = -14
Then, combine like terms:
-3x + 7 = -14
Subtract 7 from both sides:
-3x = -21
Finally, divide both sides by -3:
x = 7
Yay! We found x! x is equal to 7.
Step 4: Substitute the Value Back to Find the Other Variable
Now that we've found the value of x, we're halfway there! The next step is to substitute this value back into either of the original equations to find the value of y. It doesn't matter which equation you choose; you'll get the same answer for y either way. This flexibility is a nice feature of the substitution method. It allows us to pick the equation that looks easiest to work with, potentially saving us some time and effort. This step is all about using our newfound knowledge of x to uncover the remaining piece of the puzzle – the value of y. Let's choose wisely and complete our solution journey. Substituting the known value back into the equations allows us to see the full picture.
Let's use the first equation, y = 2x - 3.5, because it's already solved for y. Substituting x = 7, we get:
y = 2(7) - 3.5
y = 14 - 3.5
y = 10.5
Awesome! We found y too! y is equal to 10.5.
Step 5: Check Your Solution
Before we declare victory, it's always a good idea to check our solution. This is a crucial step to ensure that we haven't made any mistakes along the way. To check our solution, we simply substitute the values we found for x and y back into both of the original equations. If both equations are true, then our solution is correct! This verification process is like a safety net, catching any potential errors and giving us confidence in our answer. It also reinforces the understanding of what it means for a solution to satisfy a system of equations. Let's put our solution to the test and confirm that we've cracked the code!
Let's substitute x = 7 and y = 10.5 into both equations:
-
Equation 1: y = 2x - 3.5
10.5 = 2(7) - 3.5 10.5 = 14 - 3.5 10.5 = 10.5 (True!) -
Equation 2: x - 2y = -14
7 - 2(10.5) = -14 7 - 21 = -14 -14 = -14 (True!)
Both equations are true! This confirms that our solution is correct.
The Solution
We did it! We successfully solved the system of equations. The solution is x = 7 and y = 10.5. We can write this as an ordered pair: (7, 10.5). This ordered pair represents the point where the two lines represented by the equations intersect on a graph. Visualizing the solution in this way provides a geometric understanding of what we've accomplished algebraically. Finding this point of intersection is the essence of solving a system of equations. We've navigated the steps, conquered the algebra, and arrived at the solution! Give yourself a pat on the back!
Conclusion
So, there you have it! Solving systems of equations using the substitution method is totally achievable. Just remember to take it one step at a time, and don't be afraid to double-check your work. With practice, you'll become a system-solving superstar! Remember, the key is to break down the problem into smaller, manageable steps. This makes the entire process less daunting and more approachable. As you gain experience, you'll develop a knack for identifying the best strategies and techniques for different types of systems. Keep practicing, and you'll find yourself confidently tackling even the most challenging equations. You've got this! This step-by-step approach not only yields correct answers but also builds a deeper understanding of the underlying mathematical principles. So, embrace the challenge, practice consistently, and watch your equation-solving skills soar!
