Solving Systems Of Equations X=5 And X+y=60

Introduction

Hey guys! Let's dive into the fascinating world of systems of equations. Solving systems of equations is a fundamental concept in algebra, essential for various applications in mathematics, science, and engineering. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In this article, we'll explore a specific system and discuss how to solve it using common methods like substitution and elimination. Understanding how to solve systems of equations is crucial for anyone looking to build a strong foundation in mathematics. These skills are not just limited to textbooks; they pop up in real-world situations like balancing chemical equations, optimizing business costs, or even planning a trip. So, let's get started and break down the process step by step!

The System of Equations

Our main focus is on the system of equations presented:

1. x = 5
2. x + y = 60

This system is relatively straightforward, making it an excellent example for understanding the basic principles of solving such systems. The first equation, x = 5, immediately gives us the value of x. This is a key piece of information that simplifies the problem significantly. The second equation, x + y = 60, involves both x and y, and our goal is to find the value of y that, along with x = 5, satisfies this equation. Think of it like this: we have a puzzle where we know one piece already (x), and we need to find the other piece (y) that fits perfectly. Systems of equations like this are common in various mathematical problems, and the ability to solve them is a crucial skill. Understanding the structure of the system – recognizing that we already have the value of one variable – helps us choose the most efficient method for solving it. So, let's move on to how we can use this information to find the value of y.

Solving by Substitution

The substitution method is one of the most common and intuitive ways to solve a system of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. In our case, we already have x isolated in the first equation, x = 5. This makes the substitution method particularly easy to apply. We can take this value of x and substitute it directly into the second equation, x + y = 60. This will give us an equation with only one variable (y), which we can then solve. Imagine you're replacing a piece in a puzzle – we're taking the known value of x and plugging it into the equation to find the missing piece, y. Here’s how it works step by step:

1. Substitute x = 5 into the second equation:
   • 5 + y = 60
2. Solve for y:
   • To isolate y, we subtract 5 from both sides of the equation:
     • y = 60 - 5
     • y = 55

So, by substituting the value of x, we've found that y = 55. This method is powerful because it reduces the system to a single equation, making it much easier to solve. The substitution method is especially useful when one of the variables is already isolated or can be easily isolated. Now that we've solved the system using substitution, let's briefly consider another method: elimination.

Solving by Elimination (Optional)

While substitution was a straightforward method for this particular system, it's worth mentioning the elimination method as another powerful technique for solving systems of equations. The elimination method involves manipulating the equations so that, when added or subtracted, one of the variables is eliminated. This results in a single equation with one variable, which can then be solved. In our system:

1. x = 5
2. x + y = 60

we could technically use elimination, but it's not the most efficient approach here. However, for the sake of understanding the method, let's consider how it might work. To eliminate x, we could multiply the first equation by -1 and then add it to the second equation. This would give us:

• -x = -5
• x + y = 60

Adding these equations, we get:

• y = 55

Notice that we arrive at the same value for y as we did with substitution. However, in this case, substitution was more direct because we already had x isolated. The elimination method shines when the coefficients of one of the variables are the same or easily made the same, allowing for straightforward addition or subtraction. It’s like having different tools in a toolbox – sometimes one tool is better suited for the job than another. While elimination might not be the optimal choice here, understanding it provides a more comprehensive toolkit for solving systems of equations.

The Solution

After applying the substitution method (and briefly considering elimination), we've arrived at the solution to the system of equations:

1. x = 5
2. y = 55

This means that the values x = 5 and y = 55 satisfy both equations in the system. To verify this, we can plug these values back into the original equations:

• For the first equation, x = 5, this is clearly true.
• For the second equation, x + y = 60, we substitute the values:
  • 5 + 55 = 60
  • 60 = 60

This confirms that our solution is correct. The solution to a system of equations is the point where the lines represented by the equations intersect (if we were to graph them). In this case, it’s a single point (x = 5, y = 55). This solution is unique for this particular system, meaning there are no other values for x and y that would satisfy both equations. Understanding how to find and verify solutions is a crucial part of mastering systems of equations. It ensures that we not only find a possible answer but also confirm its accuracy. So, we've successfully found the values of x and y that make both equations true!

Real-World Applications

Solving systems of equations isn't just an abstract mathematical exercise; it has numerous real-world applications. These systems pop up in various fields, from science and engineering to economics and everyday problem-solving. Let's take a quick look at some examples to see how these skills come into play.

• Balancing Chemical Equations: In chemistry, systems of equations are used to balance chemical equations, ensuring that the number of atoms of each element is the same on both sides of the equation.
• Economics: Economists use systems of equations to model supply and demand curves, determining equilibrium prices and quantities in markets.
• Engineering: Engineers use systems of equations to analyze circuits, design structures, and solve problems involving multiple variables and constraints.
• Computer Graphics: Systems of equations are used in computer graphics to perform transformations, such as scaling, rotation, and translation of objects.
• Everyday Life: Even in everyday situations, we use the principles of solving systems of equations. For example, if you're planning a trip and need to figure out the cheapest combination of flights and hotels, you might implicitly set up and solve a system of equations.

The ability to recognize and solve systems of equations is a valuable skill that can be applied in many different contexts. It’s like having a versatile tool in your problem-solving toolkit – you might not use it every day, but when you need it, it’s incredibly useful. So, keep practicing and honing your skills in this area!

Conclusion

Alright guys, we've journeyed through the process of solving a system of equations, specifically:

1. x = 5
2. x + y = 60

We successfully used the substitution method to find the values x = 5 and y = 55. We also briefly touched on the elimination method as an alternative approach. This exercise demonstrates the fundamental principles of solving systems of equations, a crucial skill in mathematics and various real-world applications. Understanding these methods not only helps in solving mathematical problems but also enhances analytical and problem-solving skills in general. Remember, practice makes perfect! The more you work with systems of equations, the more comfortable and confident you'll become. So, keep exploring different types of systems, trying out different methods, and applying your knowledge to real-world scenarios. Whether you're balancing a budget, designing a bridge, or simply solving a puzzle, the skills you've gained here will serve you well. Keep up the great work, and happy solving!