Solving Systems Of Equations Using Elimination Method

Jun 16, 2025 by ADMIN 54 views

Solving Systems of Equations Using Elimination: A Comprehensive Guide

In the realm of mathematics, solving systems of equations is a fundamental skill with applications across various fields. One powerful technique for tackling these systems is the elimination method. This method involves manipulating the equations in a system to eliminate one variable, making it easier to solve for the other. In this comprehensive guide, we'll delve deep into the elimination method, illustrating its application through two distinct systems of equations. We'll break down each step, providing clear explanations and insights to help you master this essential technique. Whether you're a student grappling with algebra or a professional seeking to refresh your mathematical toolkit, this guide will equip you with the knowledge and confidence to solve systems of equations using elimination effectively. So, let's embark on this mathematical journey together and unlock the secrets of the elimination method.

System 1: Unveiling the Solution

Let's consider the first system of equations:

4x + 3y = 4
-2x - 3y = -8

The elimination method hinges on the strategic addition or subtraction of equations to cancel out one of the variables. In this particular system, we observe that the 'y' terms have opposite coefficients (+3 and -3). This presents a golden opportunity for direct elimination. By adding the two equations together, the 'y' terms will gracefully vanish, leaving us with a simpler equation in terms of 'x' alone.

Step-by-Step Elimination

Align and Add: We carefully align the equations, ensuring that like terms are stacked vertically. Then, we add the equations term by term:
```
(4x + 3y) + (-2x - 3y) = 4 + (-8)
```
Simplify: Combining like terms, we get:
```
2x = -4
```
Solve for x: To isolate 'x', we divide both sides of the equation by 2:
```
x = -2
```
Thus, we've successfully determined the value of 'x' to be -2. This is a significant milestone in our quest to solve the system.

Back-Substitution: Finding the Value of y

Now that we've found the value of 'x', we can substitute it back into either of the original equations to solve for 'y'. This process is known as back-substitution. Let's choose the first equation for this purpose:

4x + 3y = 4

Substitute: We replace 'x' with its value, -2:
```
4(-2) + 3y = 4
```
Simplify: Performing the multiplication, we get:
```
-8 + 3y = 4
```
Isolate y: To isolate the 'y' term, we add 8 to both sides:
```
3y = 12
```
Solve for y: Finally, we divide both sides by 3 to find the value of 'y':
```
y = 4
```
Therefore, the value of 'y' is 4. We've successfully determined the values of both 'x' and 'y' for the first system.

The Solution: A Coordinate Pair

The solution to a system of equations is typically expressed as an ordered pair (x, y). In this case, the solution to System 1 is:

(-2, 4)

This ordered pair represents the point where the lines represented by the two equations intersect on a graph. It's the unique pair of values that satisfies both equations simultaneously. Understanding the significance of this solution is crucial for grasping the essence of solving systems of equations.

System 2: A Twist in the Tale

Now, let's tackle the second system of equations:

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

In this system, a direct addition or subtraction of the equations won't immediately eliminate any variable. However, the elimination method is versatile and adaptable. We can manipulate the equations by multiplying them by suitable constants to create opposite coefficients for either 'x' or 'y'. This strategic manipulation is the key to unlocking the solution.

Strategic Multiplication: Setting the Stage for Elimination

Let's choose to eliminate 'y' in this system. To achieve this, we'll multiply the second equation by 2. This will give us a '-2y' term, which is the opposite of the '+2y' term in the first equation.

Multiply the Second Equation: Multiplying each term of the second equation by 2, we get:
```
2(2x - y) = 2(7)
4x - 2y = 14
```
Rewrite the System: Now, our system looks like this:
```
3x + 2y = 7
4x - 2y = 14
```
We've successfully transformed the system into a form where elimination is readily applicable.

Elimination: The Moment of Truth

With the 'y' terms now having opposite coefficients, we can proceed with the elimination step. We add the two equations together:

Add the Equations:
```
(3x + 2y) + (4x - 2y) = 7 + 14
```
Simplify: Combining like terms, we obtain:
```
7x = 21
```
Solve for x: Dividing both sides by 7, we find:
```
x = 3
```
Thus, the value of 'x' is 3. We're one step closer to the complete solution.

Back-Substitution: Completing the Puzzle

Now, we substitute the value of 'x' back into one of the original equations to solve for 'y'. Let's use the second equation:

2x - y = 7

Substitute: Replacing 'x' with 3, we get:
```
2(3) - y = 7
```
Simplify: Performing the multiplication:
```
6 - y = 7
```
Isolate y: Subtracting 6 from both sides:
```
-y = 1
```
Solve for y: Multiplying both sides by -1:
```
y = -1
```
Hence, the value of 'y' is -1. We've successfully determined the values of both variables for the second system.

The Solution: An Ordered Pair Representation

The solution to System 2 is the ordered pair:

(3, -1)

This point represents the intersection of the lines represented by the two equations. It's the unique solution that satisfies both equations simultaneously, highlighting the power and precision of the elimination method.

Conclusion: Mastering the Elimination Method

In this comprehensive guide, we've explored the elimination method for solving systems of equations. We've dissected two distinct systems, showcasing the method's versatility and adaptability. From direct elimination to strategic multiplication, we've covered the key techniques that empower you to tackle a wide range of systems. Remember, the elimination method is not just a mathematical tool; it's a problem-solving strategy that fosters logical thinking and analytical skills. By mastering this method, you'll not only excel in mathematics but also develop a valuable skillset applicable to various domains. So, embrace the challenge, practice diligently, and unlock the power of elimination to conquer any system of equations that comes your way.