Solving X+y=5 And X-y=1 A Step-by-Step Guide

Jun 25, 2025 by ADMIN 45 views

What is the solution to the system of equations x+y=5 and x-y=1?

Solving Systems of Equations: A Step-by-Step Guide to Finding the Solution

In mathematics, a system of equations is a set of two or more equations with 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 or curves represented by the equations intersect on a graph. Systems of equations are fundamental tools used in various fields, including mathematics, physics, engineering, economics, and computer science. They allow us to model real-world problems involving multiple constraints or relationships and find the values that satisfy all conditions.

One common type of system is a system of two linear equations with two variables, often represented as:

ax + by = c
dx + ey = f

where x and y are the variables, and a, b, c, d, e, and f are constants. To solve such a system, we need to find the values of x and y that satisfy both equations. There are several methods to solve systems of equations, each with its advantages and suitability for different types of problems. This article will explore the main methods for solving systems of equations and provide a step-by-step guide to finding the solution for the given system:

x + y = 5
x - y = 1

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, each with its strengths and weaknesses. The most common methods are:

Graphing: This method involves plotting the equations on a coordinate plane and finding the point(s) of intersection, which represent the solution(s) to the system. While visually intuitive, this method may not always yield precise solutions, especially when dealing with non-integer solutions.
Substitution: In this method, one equation is solved for one variable in terms of the other, and then this expression is substituted into the other equation. This results in a single equation with one variable, which can be easily solved. The value of the other variable can then be found by back-substitution. This method is particularly effective when one of the equations is already solved for one variable or can be easily solved.
Elimination (or Addition): This method involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, the equations are added together, eliminating one variable and resulting in a single equation with one variable. This equation can be solved, and the value of the eliminated variable can be found by back-substitution. The elimination method is advantageous when the coefficients of one variable are easily made opposites.

Graphing Method

The graphing method is a visual approach to solving systems of equations. It involves plotting each equation in the system on the same coordinate plane. The point(s) where the graphs intersect represent the solution(s) to the system, as these points satisfy all equations simultaneously. The graphing method is particularly useful for visualizing the solutions and understanding the nature of the system, such as whether there is a unique solution, infinitely many solutions, or no solution. However, it may not always provide precise solutions, especially when dealing with non-integer solutions or complex equations. To apply the graphing method, we first need to rewrite each equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This form makes it easy to plot the lines on the coordinate plane. Once the lines are plotted, we look for the point(s) of intersection. If the lines intersect at a single point, the coordinates of that point represent the unique solution to the system. If the lines are parallel and do not intersect, the system has no solution. If the lines coincide (are the same line), the system has infinitely many solutions.

Substitution Method

The substitution method is an algebraic technique for solving systems of equations. It involves solving one equation for one variable in terms of the other variable and then substituting that expression into the other equation. This process eliminates one variable, resulting in a single equation with one variable, which can be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. The substitution method is particularly effective when one of the equations is already solved for one variable or can be easily solved. It is a versatile method that can be applied to various types of systems, including linear and nonlinear systems. The key to using the substitution method successfully is to choose the equation and variable that will lead to the simplest expression after substitution. This often involves selecting an equation where one variable has a coefficient of 1 or -1, as this minimizes the need for fractions. After solving for one variable, it's crucial to substitute the value back into one of the original equations to find the value of the other variable. This step ensures that the solution satisfies both equations in the system.

Elimination Method

The elimination method, also known as the addition method, is another algebraic technique for solving systems of equations. It involves manipulating the equations so that the coefficients of one of the variables are opposites. This is typically achieved by multiplying one or both equations by a constant. Once the coefficients of one variable are opposites, the equations are added together, which eliminates that variable and results in a single equation with one variable. This equation can be easily solved, and the value of the eliminated variable can be found by back-substitution. The elimination method is particularly advantageous when the coefficients of one variable are easily made opposites, such as when they are multiples of each other or have opposite signs. It is a systematic method that can be applied to linear systems of equations with any number of variables. The key to using the elimination method effectively is to choose the appropriate multipliers to make the coefficients of one variable opposites. This may involve multiplying one or both equations by a constant. After adding the equations, it's crucial to solve for the remaining variable and then substitute the value back into one of the original equations to find the value of the eliminated variable. This ensures that the solution satisfies both equations in the system.

Solving the System of Equations: x + y = 5 and x - y = 1

Now, let's apply these methods to solve the given system of equations:

x + y = 5
x - y = 1

We will demonstrate the substitution and elimination methods to solve the system. The graphing method could also be used, but it is less precise for finding exact solutions.

1. Solving by Substitution

Step 1: Solve one equation for one variable. Let's solve the first equation for x:
```
x + y = 5
x = 5 - y
```
Step 2: Substitute the expression into the other equation. Substitute x = 5 - y into the second equation:
```
(5 - y) - y = 1
```
Step 3: Solve for the remaining variable. Simplify and solve for y:
```
5 - 2y = 1
-2y = -4
y = 2
```
Step 4: Substitute the value back to find the other variable. Substitute y = 2 into the equation x = 5 - y:
```
x = 5 - 2
x = 3
```

Therefore, the solution to the system of equations using the substitution method is x = 3 and y = 2.

2. Solving by Elimination

Step 1: Align the equations. Write the equations so that the x and y terms are aligned:
```
x + y = 5
x - y = 1
```
Step 2: Eliminate one variable. Notice that the y terms have opposite coefficients (+1 and -1). Add the two equations together:
```
(x + y) + (x - y) = 5 + 1
2x = 6
```
Step 3: Solve for the remaining variable. Divide both sides by 2 to solve for x:
```
x = 3
```
Step 4: Substitute the value back to find the other variable. Substitute x = 3 into either of the original equations. Let's use the first equation:
```
3 + y = 5
y = 2
```

Therefore, the solution to the system of equations using the elimination method is also x = 3 and y = 2.

The Solution

Both the substitution and elimination methods lead to the same solution: x = 3 and y = 2. This means that the point (3, 2) is the intersection of the two lines represented by the equations x + y = 5 and x - y = 1. To verify the solution, we can substitute these values back into the original equations:

For the first equation: 3 + 2 = 5 (True)
For the second equation: 3 - 2 = 1 (True)

Since the values satisfy both equations, we can confidently conclude that the solution to the system of equations is (3, 2). Therefore, the correct answer is C. (3,2).

Conclusion

Solving systems of equations is a fundamental skill in mathematics with applications in various fields. This article has provided a comprehensive guide to solving systems of equations, focusing on the substitution and elimination methods. We have demonstrated how to apply these methods to find the solution for the given system:

x + y = 5
x - y = 1

Both methods consistently yielded the solution x = 3 and y = 2, which corresponds to option C. (3,2). Understanding these methods and practicing their application will empower you to solve a wide range of problems involving systems of equations. Remember to always verify your solution by substituting the values back into the original equations to ensure accuracy. Mastery of solving systems of equations opens doors to tackling more complex mathematical concepts and real-world problems. Whether you are a student learning algebra or a professional applying mathematical models, the ability to solve systems of equations is an invaluable asset.