Solving 3x + 7y = 21 And 3x - Y = 3 With Substitution A Step-by-Step Guide
Solving systems of linear equations is a fundamental skill in algebra, and the substitution method is a powerful technique for finding solutions. Guys, in this article, we're going to dive deep into how to use the substitution method to solve the system of equations:
- 3x + 7y = 21
- 3x - y = 3
We'll break down each step in detail, making it super easy to follow along and master this method. So, grab your pencils and let's get started!
Understanding Systems of Equations
Before we jump into the substitution method, let's quickly recap what a system of equations actually is. Basically, a system of equations is just a set of two or more equations that involve the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph.
In our case, we have two equations:
- 3x + 7y = 21
- 3x - y = 3
Both equations involve the variables 'x' and 'y'. Our goal is to find the values of 'x' and 'y' that satisfy both equations at the same time. There are several ways to solve systems of equations, including graphing, elimination, and, of course, substitution. Today, we're focusing on the substitution method, which is particularly useful when one of the variables can be easily isolated.
The substitution method shines when we can easily single out a variable in one equation. This sets the stage for a smooth transition into solving for the remaining variable. We then bring it all home by finding the value of the variable we initially isolated. Think of it as a strategic maneuver in the world of algebra, allowing us to untangle complex equations step by step.
The cornerstone of the substitution method lies in its flexibility and applicability across a wide range of algebraic problems. Whether dealing with linear equations, quadratic equations, or even systems involving more complex functions, the underlying principle remains the same: isolate, substitute, and solve. This adaptability makes it an indispensable tool in the mathematician's toolkit, empowering us to tackle intricate problems with confidence and precision. Furthermore, mastering the substitution method not only enhances our problem-solving capabilities but also deepens our understanding of algebraic principles, paving the way for more advanced mathematical concepts and applications. So, let's embrace the power of substitution and unlock the doors to mathematical mastery!
When we're faced with a system of equations, the substitution method offers a clear pathway to finding the solution. It's a bit like piecing together a puzzle, where each step brings us closer to the final answer. By isolating one variable, we create an opportunity to express it in terms of the other, effectively reducing the complexity of the problem. This substitution allows us to transform the system into a single equation with one variable, which we can then solve using standard algebraic techniques. Once we've found the value of one variable, we can easily substitute it back into one of the original equations to find the value of the other. This process not only provides us with the solution but also reinforces our understanding of the relationships between variables within the system. So, let's embark on this journey of discovery, where each substitution brings us closer to unlocking the secrets of algebraic equations.
Step 1: Choose an Equation and Isolate a Variable
The first key step in the substitution method is to pick one of the equations and isolate one of the variables. This means we want to get either 'x' or 'y' by itself on one side of the equation. Looking at our system:
- 3x + 7y = 21
- 3x - y = 3
Equation (2) looks like a good candidate because the 'y' term has a coefficient of -1. This means it's relatively easy to isolate 'y'. Let's isolate 'y' in equation (2):
3x - y = 3
Subtract 3x from both sides:
-y = -3x + 3
Multiply both sides by -1 to get 'y' by itself:
y = 3x - 3
Great! We've now isolated 'y' in terms of 'x'. This is a crucial step because we can now substitute this expression for 'y' into the other equation.
In the realm of solving equations, the art of isolating a variable holds paramount importance. It's akin to strategically positioning a piece in a chess game, where each move sets the stage for future maneuvers. When we isolate a variable, we're essentially peeling back the layers of the equation to reveal its underlying structure. This isolation allows us to express the variable in terms of others, providing a pathway to simplification and ultimately, the solution. Whether dealing with linear equations, quadratic equations, or more complex systems, the ability to isolate variables is a fundamental skill that empowers us to unravel the mysteries of algebra. So, let's embrace the power of isolation and unlock the doors to mathematical understanding.
Isolating a variable isn't just about rearranging terms; it's about gaining a deeper understanding of the relationships within an equation. It's like dissecting a machine to see how each part contributes to the overall function. When we isolate a variable, we're essentially creating a new perspective on the equation, one that highlights the variable's dependence on the others. This newfound perspective can be incredibly valuable when solving systems of equations, as it allows us to substitute one expression for another, effectively reducing the complexity of the problem. Moreover, the act of isolating a variable hones our algebraic skills, strengthening our ability to manipulate equations and navigate the intricacies of mathematical expressions. So, let's sharpen our tools of isolation and embark on a journey of mathematical exploration.
When you're on the hunt to isolate a variable, it's like being a detective searching for the easiest clue to follow. You're looking for the variable that's most conveniently situated, the one with the fewest obstacles in its path. This might mean choosing the variable with a coefficient of 1 or -1, or perhaps the one that's already mostly alone on one side of the equation. The goal is to minimize the algebraic gymnastics required to get the variable by itself. By strategically selecting the variable to isolate, you can streamline the solving process and avoid unnecessary complications. It's a bit like choosing the right tool for the job – it makes the task at hand much smoother and more efficient. So, let's channel our inner detectives and find the easiest path to variable isolation!
Step 2: Substitute the Expression into the Other Equation
Now that we have y = 3x - 3, we can substitute this expression for 'y' into the other equation (equation 1): 3x + 7y = 21.
Replace 'y' with (3x - 3):
3x + 7(3x - 3) = 21
This substitution is super important because it eliminates one of the variables. We now have an equation with only 'x', which we can solve. This is the heart of the substitution method – transforming a system of two equations into a single equation with one variable.
Substitution in algebra is like a carefully choreographed dance, where one expression seamlessly steps in to take the place of another. It's a strategic maneuver that allows us to simplify equations and solve for unknown variables. When we substitute, we're essentially replacing one piece of the puzzle with an equivalent piece, maintaining the integrity of the equation while making it more manageable. This technique is particularly powerful when dealing with systems of equations, where substitution can transform a complex problem into a simpler one that we can easily solve. So, let's embrace the elegance of substitution and unlock the secrets hidden within algebraic expressions.
The beauty of substitution lies in its ability to bridge the gap between different expressions, allowing us to see the interconnectedness of mathematical concepts. It's like having a universal translator that can convert one form into another, revealing hidden relationships and simplifying complex problems. When we substitute, we're not just replacing symbols; we're replacing ideas, connecting them in a way that illuminates the path to the solution. This process not only enhances our problem-solving skills but also deepens our understanding of mathematical principles, fostering a sense of creativity and innovation in our approach to algebra. So, let's explore the transformative power of substitution and unlock new dimensions of mathematical insight.
When you're gearing up for substitution, think of yourself as a master chef preparing a culinary masterpiece. You've got all your ingredients laid out, and now it's time to combine them in the perfect way. The expression you're substituting is like a key ingredient that's going to add flavor to the dish. You want to make sure you add it in the right amount and at the right time. So, before you start, take a moment to visualize the substitution process. Picture the expression seamlessly integrating into the equation, transforming it into something new and exciting. This mental preparation will help you avoid mistakes and ensure a successful substitution. Now, let's get cooking!
Step 3: Solve for the Remaining Variable
Now we have the equation 3x + 7(3x - 3) = 21. Let's solve for 'x':
First, distribute the 7:
3x + 21x - 21 = 21
Combine like terms:
24x - 21 = 21
Add 21 to both sides:
24x = 42
Divide both sides by 24:
x = 42/24
Simplify the fraction:
x = 7/4
So, we've found the value of 'x'! x = 7/4. This is a major milestone. Now we just need to find the value of 'y'.
Solving for the unknown is the heart and soul of algebra, the quest that drives us through equations and expressions. It's like embarking on a treasure hunt, where each step brings us closer to the hidden prize. When we solve for a variable, we're not just finding a number; we're uncovering a piece of the puzzle, revealing a fundamental truth about the relationship between quantities. This process requires patience, persistence, and a keen eye for detail. We must carefully manipulate equations, applying the rules of algebra with precision and care. But the reward is well worth the effort, for in solving for the unknown, we gain a deeper understanding of the world around us.
The art of solving equations is like untangling a knot, where each twist and turn requires careful attention. It's a process of simplification, where we gradually peel away the layers to reveal the core relationship. We use algebraic operations as our tools, strategically applying them to isolate the variable we seek. Each step is a small victory, a step closer to the final solution. And as we progress, we gain confidence in our ability to manipulate equations, to transform them into forms that reveal their hidden truths. So, let's embrace the challenge of solving equations and unlock the power of algebra.
When you're in the thick of solving for a variable, it's easy to get caught up in the details and lose sight of the big picture. That's why it's so important to have a plan. Before you start, take a moment to map out your strategy. What steps do you need to take to isolate the variable? What operations will you need to perform? By having a clear plan in mind, you can avoid mistakes and stay on track. It's like having a GPS for your algebraic journey. So, let's chart our course and navigate our way to the solution!
Step 4: Substitute the Value Back to Find the Other Variable
Now that we know x = 7/4, we can substitute this value back into either equation (1) or equation (2) to find 'y'. However, it's usually easier to substitute into the equation where we already isolated 'y', which is y = 3x - 3.
Substitute x = 7/4 into y = 3x - 3:
y = 3(7/4) - 3
y = 21/4 - 3
To subtract 3, we need to express it as a fraction with a denominator of 4:
y = 21/4 - 12/4
y = 9/4
So, we've found the value of 'y'! y = 9/4.
Substitution, the encore! Just when you thought the show was over, substitution makes a triumphant return in the final act of solving for the other variable. It's like bringing in a star player off the bench to seal the victory. We've already found the value of one variable, and now it's time to use that knowledge to unlock the mystery of the other. By substituting the known value back into one of the original equations, we create a new equation with only one unknown. This equation is now ripe for the solving, and with a few algebraic maneuvers, we can unveil the value of the remaining variable. So, let's give substitution its moment in the spotlight and bring this mathematical performance to a resounding conclusion!
The act of substituting back isn't just about plugging in a number; it's about completing the puzzle. It's the moment when all the pieces come together, revealing the full picture. We've spent time isolating variables, simplifying expressions, and solving equations, and now we get to reap the rewards of our efforts. By substituting the known value back into the equation, we're not just finding the other variable; we're confirming the validity of our solution. It's a moment of satisfaction, a testament to the power of algebra and our ability to navigate its complexities. So, let's savor this final step and celebrate the successful completion of our mathematical journey!
When you're substituting a value back to find the other variable, it's like returning home after a long journey. You're going back to where you started, but now you have something new to bring with you: the value of one of the variables. This new information is like a key that unlocks the door to the final solution. You can use it to find the value of the other variable, completing the puzzle and bringing everything into focus. So, let's make our way back home, armed with our newfound knowledge, and unlock the secrets that await us!
Step 5: Write the Solution as an Ordered Pair
We've found that x = 7/4 and y = 9/4. The solution to a system of equations is usually written as an ordered pair (x, y). So, our solution is:
(7/4, 9/4)
This means that the point (7/4, 9/4) is the intersection point of the two lines represented by the equations 3x + 7y = 21 and 3x - y = 3. We've successfully solved the system using the substitution method!
Writing the solution as an ordered pair is like putting the final brushstroke on a masterpiece. It's the culmination of all our hard work, the moment when we present our solution to the world. The ordered pair is a concise and elegant way to express the values of the variables that satisfy the system of equations. It's a symbol of completion, a testament to our algebraic prowess. So, let's take pride in our accomplishment and present our solution with confidence!
The ordered pair is more than just a set of numbers; it's a point in space, a meeting place where two lines intersect. It's a visual representation of the solution, a tangible manifestation of the algebraic concept. When we write the solution as an ordered pair, we're not just stating the values of the variables; we're describing a location, a point of equilibrium in the mathematical landscape. This connection between algebra and geometry is a powerful one, and it highlights the beauty and interconnectedness of mathematical ideas. So, let's embrace the visual aspect of the ordered pair and appreciate its significance in the world of mathematics!
When you're writing the solution as an ordered pair, think of it as signing your name on a work of art. It's your personal stamp of approval, a declaration that you've successfully solved the problem. The ordered pair is a concise and professional way to present your solution, and it shows that you understand the conventions of mathematical notation. So, let's put the finishing touches on our masterpiece and sign our name with pride!
Conclusion
Guys, we've successfully used the substitution method to solve the system of equations 3x + 7y = 21 and 3x - y = 3. We found that the solution is (7/4, 9/4). Remember, the key steps are:
- Isolate a variable in one equation.
- Substitute the expression into the other equation.
- Solve for the remaining variable.
- Substitute the value back to find the other variable.
- Write the solution as an ordered pair.
Practice makes perfect, so try solving more systems of equations using the substitution method. You'll get the hang of it in no time! Keep up the great work!
