Solving System Of Equations Y=5x+2 And 3x=-y+10
Introduction
In the realm of mathematics, solving systems of equations is a fundamental skill. These systems represent situations where multiple variables are related through multiple equations, and the goal is to find values for the variables that satisfy all equations simultaneously. This article dives deep into solving the specific system of equations: y = 5x + 2 and 3x = -y + 10. We'll explore various methods, analyze the solution, and discuss the underlying concepts. Understanding how to solve systems of equations is crucial not only in mathematics but also in various fields like physics, engineering, and economics, where models often involve multiple interrelated variables. Whether you're a student tackling algebra problems or a professional applying mathematical models, mastering these techniques will prove invaluable. The beauty of mathematics lies in its ability to provide precise solutions to complex problems, and solving systems of equations is a prime example of this. So, let's embark on this journey of mathematical discovery and unlock the secrets hidden within these equations.
Understanding the Problem
Before we jump into solving, let's make sure we understand what the problem is asking. We're given two equations:
- y = 5x + 2
- 3x = -y + 10
Each equation represents a straight line when graphed on a coordinate plane. The solution to the system of equations is the point (x, y) where these two lines intersect. This point represents the unique pair of values for x and y that satisfy both equations. Think of it as finding the exact spot where two paths cross – the x and y coordinates of that spot are our solution. There are several ways to find this intersection point, and we will explore the most common and efficient methods. These methods include substitution, elimination, and graphing. Each method offers a different approach to manipulating the equations and isolating the variables. Understanding the underlying principles of each method will allow you to choose the most appropriate technique for a given system of equations. Ultimately, the goal is to find the values of x and y that make both equations true, providing a complete and accurate solution to the system. By understanding the graphical representation and the algebraic techniques, we can confidently tackle a wide range of problems involving systems of equations.
Method 1 Substitution Method
The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable, which we can then solve. Let's apply this method to our system:
- y = 5x + 2
- 3x = -y + 10
Notice that the first equation is already solved for y. This makes it a perfect candidate for substitution. We can substitute the expression 5x + 2 for y in the second equation:
3x = -(5x + 2) + 10
Now we have an equation with only x. Let's simplify and solve for x:
3x = -5x - 2 + 10 3x = -5x + 8 8x = 8 x = 1
Great! We've found the value of x. Now we can substitute this value back into either of the original equations to find y. Let's use the first equation, y = 5x + 2:
y = 5(1) + 2 y = 5 + 2 y = 7
So, using the substitution method, we've found the solution: x = 1 and y = 7. This means the point of intersection of the two lines is (1, 7). The substitution method is particularly useful when one of the equations is already solved for a variable or can be easily solved. By systematically substituting expressions, we can reduce the complexity of the system and arrive at the solution with confidence. This method highlights the power of algebraic manipulation in simplifying mathematical problems.
Method 2 Elimination Method
The elimination method, also known as the addition method, is another effective technique for solving systems of equations. This method involves manipulating the equations so that when they are added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which we can solve. Let's apply the elimination method to our system:
- y = 5x + 2
- 3x = -y + 10
First, we need to rearrange the equations so that the x and y terms are aligned. Let's rewrite the first equation as:
-5x + y = 2
And the second equation as:
3x + y = 10
Now, we want to eliminate one of the variables. Notice that the y terms have the same coefficient (1). To eliminate y, we can multiply the first equation by -1:
(-1)(-5x + y) = (-1)(2) 5x - y = -2
Now we have the modified system:
- 5x - y = -2
- 3x + y = 10
Now we can add the two equations together:
(5x - y) + (3x + y) = -2 + 10 8x = 8 x = 1
We've found the value of x! Now we can substitute this value back into either of the original equations to find y. Let's use the first original equation, y = 5x + 2:
y = 5(1) + 2 y = 5 + 2 y = 7
So, using the elimination method, we've again found the solution: x = 1 and y = 7. This confirms our previous result using the substitution method. The elimination method is particularly useful when the coefficients of one of the variables are the same or easily made the same by multiplication. By strategically adding or subtracting equations, we can eliminate variables and simplify the system. This method provides a systematic way to solve systems of equations and offers a valuable alternative to the substitution method.
Method 3 Graphing
Graphing provides a visual approach to solving systems of equations. Each equation in the system represents a line, and the solution to the system is the point where the lines intersect. Let's graph our system of equations:
- y = 5x + 2
- 3x = -y + 10
To graph the first equation, y = 5x + 2, we can identify the slope and y-intercept. The slope is 5, and the y-intercept is 2. This means the line crosses the y-axis at the point (0, 2), and for every 1 unit we move to the right, we move 5 units up. Plotting a few points and connecting them gives us the graph of the first line.
For the second equation, 3x = -y + 10, let's rewrite it in slope-intercept form (y = mx + b):
y = -3x + 10
Now we can see that the slope is -3, and the y-intercept is 10. This line crosses the y-axis at the point (0, 10), and for every 1 unit we move to the right, we move 3 units down. Plotting a few points and connecting them gives us the graph of the second line.
By graphing both lines on the same coordinate plane, we can visually identify the point of intersection. Looking at the graph, we can see that the lines intersect at the point (1, 7). This confirms our solutions obtained using the substitution and elimination methods. Graphing provides a valuable visual confirmation of the algebraic solutions. It allows us to see the relationship between the equations and their corresponding lines. While graphing may not always provide the most precise solution, especially when dealing with non-integer values, it offers a powerful tool for understanding the behavior of systems of equations. In addition, graphing calculators and software can be used to accurately graph equations and find intersection points, making this method even more effective.
The Solution and its Verification
We've used three different methods – substitution, elimination, and graphing – to solve the system of equations:
- y = 5x + 2
- 3x = -y + 10
Each method has consistently led us to the same solution: x = 1 and y = 7. This means the point of intersection of the two lines represented by these equations is (1, 7). But how can we be absolutely sure that this is the correct solution? The answer lies in verification. To verify our solution, we substitute the values x = 1 and y = 7 back into the original equations and check if they hold true.
Let's substitute into the first equation:
y = 5x + 2 7 = 5(1) + 2 7 = 5 + 2 7 = 7 (True)
Now let's substitute into the second equation:
3x = -y + 10 3(1) = -7 + 10 3 = 3 (True)
Since the values x = 1 and y = 7 satisfy both equations, we can confidently conclude that this is the correct solution to the system. This verification process is crucial in mathematics. It provides a check against errors and ensures the accuracy of the solution. By substituting the solution back into the original equations, we confirm that the values we've found indeed satisfy the relationships defined by the equations. This not only reinforces our understanding of the solution process but also builds confidence in the result.
Conclusion
In this comprehensive guide, we've explored the process of solving the system of equations:
- y = 5x + 2
- 3x = -y + 10
We've employed three distinct methods – substitution, elimination, and graphing – to arrive at the solution. Each method provides a unique perspective and approach to solving systems of equations, highlighting the versatility of mathematical tools. The consistent result obtained through these methods, x = 1 and y = 7, reinforces the accuracy and reliability of our solution. We further validated our findings by substituting these values back into the original equations, confirming that they satisfy both conditions.
Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. From physics and engineering to economics and computer science, the ability to model and solve systems of equations is essential for understanding and solving real-world problems. The techniques we've discussed – substitution, elimination, and graphing – provide a solid foundation for tackling a variety of systems of equations. Mastering these methods not only enhances your mathematical proficiency but also equips you with valuable problem-solving skills applicable across diverse fields. The key to success lies in understanding the underlying principles of each method and choosing the most appropriate technique for a given problem. With practice and perseverance, you can confidently navigate the world of systems of equations and unlock their power to model and solve complex scenarios.
