Solving Systems Of Equations Elimination Method Step-by-Step Guide

Jul 19, 2025 by Jeany 67 views

Solve Systems of Equations Using the Elimination Method

In mathematics, a system of equations is a set of two or more equations containing the same variables. A solution to a system of equations is a set of values for the variables that makes all the equations true. There are several methods for solving systems of equations, including substitution, graphing, and elimination. In this article, we will focus on the elimination method, a powerful technique that allows us to solve systems of equations by strategically adding or subtracting the equations to eliminate one variable.

The elimination method, also known as the addition method, is a technique used to solve systems of linear equations. The basic idea behind the elimination method is to manipulate the equations in such a way that when they are added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which can be easily solved. Once we have the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

Understanding the Elimination Method

The elimination method is particularly useful when dealing with systems of equations where the coefficients of one of the variables are opposites or can be easily made opposites by multiplying one or both equations by a constant. This allows us to add the equations together and eliminate that variable, simplifying the system and making it easier to solve.

To effectively utilize the elimination method, follow these key steps:

Align the variables: Ensure that the variables in both equations are aligned in columns. This means that the x terms should be above each other, the y terms above each other, and the constants on the right side of the equals sign.
Create opposite coefficients: Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 2x and -2x). This is the crucial step that allows for elimination.
Add the equations: Add the equations together vertically. The variable with opposite coefficients will be eliminated, leaving you with a single equation in one variable.
Solve for the remaining variable: Solve the resulting equation for the remaining variable.
Substitute back: Substitute the value you found in step 4 back into either of the original equations and solve for the other variable.
Check your solution: Substitute the values you found for both variables into both original equations to ensure they satisfy both equations. This verifies the correctness of your solution.

Step-by-Step Example: Solving a System of Equations by Elimination

Let's illustrate the elimination method with a step-by-step example. Consider the following system of equations:

2x + y = 7
3x - y = 8

Align the variables: The variables are already aligned in columns.
Create opposite coefficients: Notice that the y coefficients are already opposites (+y and -y). No multiplication is needed in this case.

Add the equations: Add the equations together:

2x + y = 7
3x - y = 8
---------
5x     = 15

Solve for the remaining variable: Divide both sides by 5 to solve for x:
```
5x = 15
x = 3
```
Substitute back: Substitute x = 3 into either of the original equations. Let's use the first equation:
```
2(3) + y = 7
6 + y = 7
y = 1
```
Check your solution: Substitute x = 3 and y = 1 into both original equations:
```
2(3) + 1 = 7  (True)
3(3) - 1 = 8  (True)
```

The solution to the system of equations is x = 3 and y = 1, which can be written as the ordered pair (3, 1).

Solving the Given System of Equations: A Detailed Walkthrough

Now, let's apply the elimination method to the specific system of equations provided:

5x - y = 39
x + 4y = 12

Our goal is to find the values of x and y that satisfy both equations simultaneously. We'll achieve this by strategically manipulating the equations to eliminate one variable, allowing us to solve for the other.

1. Align the Variables

The equations are already neatly aligned, with the x terms, y terms, and constants in their respective columns. This is a crucial first step, ensuring we can perform the elimination process correctly.

5x - y = 39
x + 4y = 12

2. Create Opposite Coefficients

To eliminate a variable, we need its coefficients in both equations to be opposites (e.g., 2 and -2). Looking at the equations, we can easily achieve this for the y variable. Notice that the first equation has a -y term, while the second has a +4y term. To make these coefficients opposites, we can multiply the first equation by 4.

Multiplying the entire first equation by 4, we get:

4 * (5x - y) = 4 * 39
20x - 4y = 156

Now our system of equations looks like this:

20x - 4y = 156
x + 4y = 12

3. Add the Equations

With the y coefficients as opposites (-4y and +4y), we can now add the two equations together. This will eliminate the y variable, leaving us with an equation in just x.

Adding the equations vertically:

20x - 4y = 156
 x + 4y = 12
----------------
21x       = 168

4. Solve for the Remaining Variable

We now have a simple equation: 21x = 168. To solve for x, we divide both sides of the equation by 21:

21x / 21 = 168 / 21
x = 8

So, we've found that x = 8.

5. Substitute Back

Now that we know the value of x, we can substitute it back into either of the original equations to solve for y. Let's use the second equation, x + 4y = 12, as it appears simpler.

Substituting x = 8 into the equation:

8 + 4y = 12

To solve for y, we first subtract 8 from both sides:

4y = 12 - 8
4y = 4

Then, we divide both sides by 4:

4y / 4 = 4 / 4
y = 1

So, we've found that y = 1.

6. Check Your Solution

To ensure our solution is correct, we need to substitute the values of x and y back into both of the original equations. If both equations hold true, our solution is verified.

Let's check the first equation, 5x - y = 39:

5(8) - 1 = 39
40 - 1 = 39
39 = 39  (True)

Now, let's check the second equation, x + 4y = 12:

8 + 4(1) = 12
8 + 4 = 12
12 = 12  (True)

Since both equations hold true, our solution is correct.

The Solution Set

The solution to the system of equations is x = 8 and y = 1. We express this as an ordered pair (x, y), which in this case is (8, 1).

Therefore, the solution set is {(8, 1)}.

Advanced Techniques and Considerations

While the basic elimination method is straightforward, there are some advanced techniques and considerations to keep in mind when solving more complex systems of equations.

Multiplying both equations: Sometimes, you may need to multiply both equations by different constants to create opposite coefficients for one of the variables. This is particularly useful when the coefficients don't have a simple common multiple.
Fractions and decimals: If your equations contain fractions or decimals, it's often helpful to clear them before applying the elimination method. Multiply the entire equation by the least common denominator (for fractions) or a power of 10 (for decimals) to eliminate these terms.
No solution or infinite solutions: In some cases, a system of equations may have no solution or infinite solutions. If, during the elimination process, you arrive at a contradiction (e.g., 0 = 5), the system has no solution. If you arrive at an identity (e.g., 0 = 0), the system has infinite solutions.
Systems with three or more variables: The elimination method can be extended to solve systems of equations with three or more variables. The basic idea is to eliminate one variable at a time until you are left with a system of equations that can be easily solved.

Applications of the Elimination Method

The elimination method is a fundamental tool in mathematics and has numerous applications in various fields, including:

Science and engineering: Solving systems of equations is crucial in many scientific and engineering applications, such as circuit analysis, structural analysis, and chemical reactions.
Economics and finance: Systems of equations are used to model economic systems, financial markets, and investment strategies.
Computer science: The elimination method is used in computer graphics, optimization algorithms, and data analysis.
Everyday life: Systems of equations can be used to solve practical problems, such as determining the cost of items, calculating mixtures, and planning budgets.

Conclusion

The elimination method is a powerful and versatile technique for solving systems of linear equations. By strategically adding or subtracting equations, we can eliminate variables and simplify the system, making it easier to find the solution. Mastering the elimination method is essential for anyone studying mathematics, science, or engineering. With practice and understanding, you can confidently tackle a wide range of problems involving systems of equations. Remember to align variables, create opposite coefficients, add equations, solve for remaining variables, substitute back, and check your solution. By following these steps, you can successfully solve systems of equations using the elimination method.

This method provides a systematic approach to solving systems of equations, making it a valuable tool in various mathematical and real-world applications. By understanding and practicing the elimination method, you can enhance your problem-solving skills and gain a deeper understanding of linear algebra concepts.