Solving Systems Of Equations 4x + Y = 10 And X = 10 - Y Step-by-Step Guide
Hey guys! Today, we're diving into a classic math problem: solving a system of equations. Specifically, we're going to tackle the system:
- 4x + y = 10
- x = 10 - y
This kind of problem pops up everywhere, from basic algebra to real-world applications, so mastering it is super important. We'll break down the steps, explain the reasoning, and make sure you've got a solid understanding of how to solve these types of problems. Let's get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what we're actually trying to do. A system of equations is just a set of two or more equations that we're trying to solve simultaneously. This means we're looking for values for our variables (in this case, x and y) that make all the equations true at the same time. Think of it like a puzzle where all the pieces have to fit together perfectly.
In our case, we have two equations and two unknowns (x and y). This is a common scenario, and it usually means we can find a unique solution. If we had more unknowns than equations, we might have infinitely many solutions or no solutions at all. But don't worry about that for now! We're focusing on the nice, solvable case.
The two equations we're working with are:
- 4x + y = 10 This is a linear equation, which means if we were to graph it, it would form a straight line. The coefficients (4 and 1) tell us the slope and intercepts of the line.
- x = 10 - y This is also a linear equation, but it's written in a slightly different form. It tells us that x is equal to 10 minus y. This form is actually quite helpful for one of the solving methods we'll use.
Our goal is to find the specific values of x and y that satisfy both of these equations. There are a few different ways we can do this, and we'll explore two of the most common methods: substitution and elimination.
Diving Deep into Solving with Substitution
The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation in a single variable, which we can then easily solve. Once we've found the value of one variable, we can plug it back into either of the original equations to find the value of the other variable. It's like a chain reaction – solve for one, then use that to solve for the other!
Looking at our system of equations:
- 4x + y = 10
- x = 10 - y
We notice that the second equation, x = 10 - y, is already solved for x. This makes substitution a very natural choice for this problem. We can directly substitute the expression "10 - y" for x in the first equation. This is the key move in the substitution method – replacing a variable with its equivalent expression.
So, let's do it! We take the first equation, 4x + y = 10, and replace x with (10 - y):
4(10 - y) + y = 10
Now we have a single equation with just one variable, y. This is great progress! We've eliminated x and can now focus on solving for y. This is the power of substitution – it simplifies the problem by reducing the number of variables.
Next, we need to simplify and solve this equation. We start by distributing the 4:
40 - 4y + y = 10
Now we combine like terms (the y terms):
40 - 3y = 10
Our goal is to isolate y, so we subtract 40 from both sides of the equation:
-3y = 10 - 40
-3y = -30
Finally, we divide both sides by -3 to solve for y:
y = -30 / -3
y = 10
Awesome! We've found the value of y: y = 10. But we're not done yet. We still need to find the value of x. This is where the second part of the substitution method comes in – plugging the value we just found back into one of the original equations.
We can use either of the original equations, but the second one, x = 10 - y, looks easier to work with. So, we substitute y = 10 into this equation:
x = 10 - 10
x = 0
And there we have it! We've found the value of x: x = 0. So, the solution to the system of equations is x = 0 and y = 10. These are the values that satisfy both equations simultaneously.
But before we declare victory, it's always a good idea to check our answer. We can do this by plugging our values for x and y back into both original equations to make sure they hold true.
Let's check the first equation, 4x + y = 10:
4(0) + 10 = 10
0 + 10 = 10
10 = 10
This equation checks out! Now let's check the second equation, x = 10 - y:
0 = 10 - 10
0 = 0
This equation also checks out! Since our solution satisfies both equations, we can be confident that we've found the correct answer.
Exploring the Elimination Method
Now, let's explore another powerful technique for solving systems of equations: the elimination method (sometimes called the addition method). This method focuses on eliminating one of the variables by manipulating the equations so that when you add them together, one of the variables cancels out. It's like strategically combining the equations to make one of the variables disappear!
The key idea behind the elimination method is that you can multiply an entire equation by a constant without changing its solution. This allows us to create coefficients that are opposites for one of the variables. When we add the equations, those terms will cancel out, leaving us with a single equation in a single variable. This is similar to the substitution method in that it aims to reduce the problem to a simpler form.
Let's revisit our system of equations:
- 4x + y = 10
- x = 10 - y
To use the elimination method effectively, it's helpful to rewrite the equations in a standard form, with the x and y terms on the same side of the equation. So, let's rewrite the second equation:
x = 10 - y becomes x + y = 10
Now our system looks like this:
- 4x + y = 10
- x + y = 10
Looking at these equations, we notice that the y terms have the same coefficient (1). This is a good starting point for elimination. To eliminate the y terms, we can multiply one of the equations by -1. This will change the sign of the y term, making it the opposite of the y term in the other equation.
Let's multiply the second equation by -1:
-1(x + y) = -1(10)
-x - y = -10
Now our system looks like this:
- 4x + y = 10
- -x - y = -10
Now comes the magic step: we add the two equations together. When we add the left-hand sides and the right-hand sides, the y terms cancel out:
(4x + y) + (-x - y) = 10 + (-10)
4x + y - x - y = 0
3x = 0
We've eliminated y and are left with a simple equation in x. This is exactly what we wanted! Now we can easily solve for x:
x = 0 / 3
x = 0
Great! We've found the value of x: x = 0. Now we need to find the value of y. Just like in the substitution method, we can plug the value we just found back into either of the original equations.
Let's use the second equation in its rewritten form, x + y = 10, since it looks simpler:
0 + y = 10
y = 10
And there we have it! We've found the value of y: y = 10. So, the solution to the system of equations, using the elimination method, is also x = 0 and y = 10. This matches the solution we found using the substitution method, which is a good sign that we're on the right track.
As always, let's check our answer by plugging our values for x and y back into the original equations:
First equation, 4x + y = 10:
4(0) + 10 = 10
0 + 10 = 10
10 = 10
This equation checks out!
Second equation, x = 10 - y:
0 = 10 - 10
0 = 0
This equation also checks out! Since our solution satisfies both equations, we can be confident that we've solved the system correctly using the elimination method.
Choosing the Right Method
So, we've seen two different methods for solving the same system of equations: substitution and elimination. You might be wondering, which method is better? The truth is, there's no single "best" method. The most efficient method often depends on the specific equations you're dealing with. Some systems are naturally suited for substitution, while others are easier to solve using elimination.
Substitution tends to be a good choice when one of the equations is already solved for one of the variables, or when it's easy to isolate one variable in one of the equations. In our example, the equation x = 10 - y made substitution a very natural choice because x was already isolated. It is very useful in more complex math and real life situation problems.
Elimination, on the other hand, shines when the coefficients of one of the variables are the same or easily made opposites. In our example, the y terms had the same coefficient, which made elimination a straightforward option. In particular, if you need to deal with more complicated equations, try to use this way.
Ultimately, the best way to decide which method to use is to look at the equations and see which method seems like it will involve the least amount of work. And, of course, practice makes perfect! The more you solve systems of equations, the better you'll become at recognizing which method is the most efficient for a given problem.
Key Takeaways
- A system of equations is a set of two or more equations that we're trying to solve simultaneously.
- The solution to a system of equations is the set of values for the variables that satisfy all the equations in the system.
- The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
- The elimination method involves manipulating the equations so that when you add them together, one of the variables cancels out.
- The best method to use depends on the specific equations in the system. Look for equations that are already solved for a variable (suggesting substitution) or variables with the same or easily made opposite coefficients (suggesting elimination).
- Always check your solution by plugging the values back into the original equations.
Solving systems of equations is a fundamental skill in algebra and beyond. By mastering these techniques, you'll be well-equipped to tackle a wide range of math problems and real-world applications. Keep practicing, and you'll become a system-solving pro in no time!
Real-World Applications
The skills you've learned today aren't just for textbooks and exams. Systems of equations pop up in all sorts of real-world situations! Let's explore a few examples to see how this math can be used in practical ways.
-
Mixing Solutions in Chemistry: Imagine you're a chemist and you need to create a specific solution with a certain concentration of acid. You have two stock solutions with different concentrations. Systems of equations can help you determine how much of each stock solution to mix to get the desired concentration and volume.
-
Supply and Demand in Economics: Economists use systems of equations to model the supply and demand of goods and services. The point where the supply and demand curves intersect represents the equilibrium price and quantity in the market. This is a fundamental concept in economics.
-
Navigation and GPS: Global Positioning System (GPS) technology relies on systems of equations to determine your location. GPS satellites transmit signals that your receiver uses to calculate your distance from each satellite. These distances are then used in a system of equations to pinpoint your exact location on Earth.
-
Circuit Analysis in Electrical Engineering: Electrical engineers use systems of equations to analyze circuits. They can use Kirchhoff's laws, which describe the flow of current and voltage in a circuit, to set up a system of equations that can be solved to find the currents and voltages in different parts of the circuit.
-
Financial Planning: You can use systems of equations to model your financial goals and plan your savings and investments. For example, you might use a system of equations to determine how much you need to save each month to reach a certain retirement goal, taking into account factors like interest rates and inflation.
These are just a few examples, but they illustrate the power and versatility of systems of equations. Whether you're mixing chemicals in a lab, analyzing economic trends, or navigating with GPS, the skills you've learned today can help you solve real-world problems.
Conclusion
We've covered a lot of ground in this article! We started by understanding what a system of equations is and why it's important. We then dove into two powerful methods for solving systems: substitution and elimination. We learned how to apply each method, how to choose the right method for a given problem, and how to check our answers. Finally, we explored some real-world applications of systems of equations to see how these skills can be used in practical situations. It is useful in chemistry, economy, financial planning, circuit analysis, and navigation.
Remember, the key to mastering systems of equations is practice. Work through lots of examples, try different methods, and don't be afraid to make mistakes. Every mistake is a learning opportunity! The more you practice, the more comfortable and confident you'll become in your ability to solve these types of problems.
So, guys, keep practicing, keep exploring, and keep applying your math skills to the world around you. You've got this!
