Solving Systems Of Equations Find The Solution For X - 2y = 15 And 2x + 4y = -18

Jul 11, 2025 by ADMIN 81 views

Solving Systems of Equations A Step-by-Step Guide

In the world of mathematics, solving systems of equations is a fundamental skill. It's a process that allows us to find the values of multiple variables that satisfy a set of equations simultaneously. Systems of equations pop up in various real-world scenarios, from determining the optimal mix of ingredients in a recipe to calculating the trajectory of a rocket. This article dives into a specific system of equations problem, walking you through the solution step-by-step. So, if you are dealing with a system of equations, this guide is designed to equip you with the knowledge and confidence to tackle similar challenges. Whether you are a student grappling with algebra or simply someone looking to brush up on their math skills, you've come to the right place. Let's embark on this mathematical journey together and unlock the secrets of solving systems of equations!

The Problem

We are presented with the following system of equations:

x - 2y = 15
2x + 4y = -18

Our mission is to find the values of x and y that make both equations true. We have four possible solutions to consider:

A. x = 1, y = -6 B. x = 1, y = -7 C. x = 3, y = -6 D. x = 3, y = -7

Methods for Solving Systems of Equations

Before we dive into the solution, let's briefly discuss the common methods used to solve systems of equations.

Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved easily.
Elimination Method: The elimination method focuses on eliminating one variable by adding or subtracting multiples of the equations. The goal is to create coefficients for one variable that are opposites, so that when the equations are added, that variable is eliminated.
Graphing Method: Graphing involves plotting the equations on a coordinate plane. The solution to the system is the point(s) where the lines intersect. This method is visually intuitive but may not be precise for non-integer solutions.

For this particular problem, we will use the elimination method, as it offers a straightforward approach to solving this system.

Step-by-Step Solution Using the Elimination Method

Multiply the First Equation by 2:

To prepare for elimination, we want the coefficients of either x or y to be opposites. Let's multiply the first equation by 2:
```
2 * (x - 2y) = 2 * 15
2x - 4y = 30
```
Now our system of equations looks like this:
```
2x - 4y = 30
2x + 4y = -18
```
Add the Modified First Equation to the Second Equation:

Notice that the coefficients of y are now opposites (-4 and +4). Adding the equations will eliminate y:
```
(2x - 4y) + (2x + 4y) = 30 + (-18)
4x = 12
```
Solve for x:

Divide both sides of the equation by 4 to isolate x:
```
4x / 4 = 12 / 4
x = 3
```
We have found that x = 3.
Substitute the Value of x into One of the Original Equations to Solve for y:

Let's use the first original equation:
```
x - 2y = 15
```
Substitute x = 3:
```
3 - 2y = 15
```
Solve for y:

Subtract 3 from both sides:
```
-2y = 12
```
Divide both sides by -2:
```
y = -6
```
We have found that y = -6.
The Solution:

Therefore, the solution to the system of equations is x = 3 and y = -6.

Verifying the Solution

It's always a good practice to verify our solution by substituting the values of x and y back into the original equations to ensure they hold true.

First Equation:

x - 2y = 15
3 - 2(-6) = 15
3 + 12 = 15
15 = 15  (True)

Second Equation:

2x + 4y = -18
2(3) + 4(-6) = -18
6 - 24 = -18
-18 = -18  (True)

Since the values x = 3 and y = -6 satisfy both equations, our solution is correct.

The Correct Answer

Looking back at the options provided, we can see that the correct answer is:

C. x = 3, y = -6

Conclusion

In this article, we successfully solved a system of equations using the elimination method. We walked through each step, from setting up the equations to verifying the solution. Understanding how to solve systems of equations is a valuable skill in mathematics and has practical applications in various fields. By mastering this concept, you'll be well-equipped to tackle more complex mathematical problems. Remember, practice makes perfect, so keep honing your skills and exploring the fascinating world of mathematics!

Key Takeaways

Systems of equations involve finding the values of multiple variables that satisfy a set of equations.
The elimination method is a powerful technique for solving systems of equations by eliminating one variable.
Always verify your solution by substituting the values back into the original equations.
Understanding the concept of solving systems of equations is useful for many problems.

Practice Problems

To solidify your understanding, try solving these systems of equations on your own:

```
3x + y = 7
x - y = 1
```
```
2x + 3y = 8
x - 2y = -3
```

Feel free to share your solutions and discuss the methods you used in the comments below.

Further Exploration

If you're interested in delving deeper into the world of systems of equations, consider exploring these topics:

Linear Equations: Understanding the properties of linear equations is crucial for solving systems of equations.
Matrices: Systems of equations can be represented and solved using matrices.
Applications of Systems of Equations: Explore how systems of equations are used in real-world scenarios, such as economics, engineering, and computer science.

By continuing your mathematical journey, you'll unlock new levels of understanding and appreciation for the power of mathematics.