Solving Systems Of Equations: A Step-by-Step Guide

Jul 9, 2025 by ADMIN 51 views

Solving a System of Equations: A Comprehensive Guide

In mathematics, a system of equations is a set of two or more equations containing the same variables. Solving a system of equations means finding the values for the variables that satisfy all equations simultaneously. There are several methods to solve systems of equations, including substitution, elimination, and matrix methods. In this comprehensive guide, we will explore the elimination method to solve the given system of equations. We'll break down each step, ensuring a clear understanding of the process. Mastering this technique will significantly enhance your problem-solving skills in algebra and beyond.

Understanding the System of Equations

Before diving into the solution, let's clearly define the system of equations we aim to solve. The given system is:

-7x - 5y = 3
4x + 9y = -14

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that make both equations true. A solution to a system of equations is an ordered pair (x, y) that satisfies each equation in the system. Graphically, the solution represents the point of intersection of the lines represented by the equations. If the lines are parallel, there is no solution; if they are the same line, there are infinitely many solutions. Understanding these fundamental concepts is crucial for tackling more complex systems of equations. The equations given represent straight lines on a coordinate plane, and the point where these lines intersect gives us the solution to the system. If the lines do not intersect, it means there is no solution that satisfies both equations simultaneously. This can happen when the lines are parallel. On the other hand, if the two equations represent the same line, there are infinitely many solutions, as every point on the line satisfies both equations.

Method: Elimination Method

The elimination method involves manipulating the equations to eliminate one variable, allowing us to solve for the other. We achieve this by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When we add the equations, this variable cancels out, leaving us with a single equation in one variable. This resulting equation can be easily solved, and then we can substitute the value back into one of the original equations to find the value of the other variable. The elimination method is particularly useful when the equations are in standard form (Ax + By = C), as it provides a systematic way to eliminate one variable. It is also beneficial when dealing with systems of equations where substitution might lead to cumbersome fractions or complex expressions. By carefully choosing the multipliers, we can simplify the process and reduce the chances of making errors. This method's efficiency and clarity make it a favorite among students and mathematicians alike for solving linear systems of equations.

Step-by-step Solution

Multiply the equations to make the coefficients of one variable opposites.
- We can eliminate x by multiplying the first equation by 4 and the second equation by 7:
```
4(-7x - 5y) = 4(3)  =>  -28x - 20y = 12
7(4x + 9y) = 7(-14) =>  28x + 63y = -98
```
- By multiplying the first equation by 4 and the second equation by 7, we've created a situation where the coefficients of x are -28 and 28, respectively. This sets the stage for eliminating x when we add the equations together. The key to this step is choosing multipliers that will make the coefficients of one variable opposites. This strategic move allows us to simplify the system and move closer to finding a solution. The multiplied equations now form a new system that is equivalent to the original but is structured to allow for the elimination of a variable.
Add the equations to eliminate the variable.
- Adding the modified equations:
```
(-28x - 20y) + (28x + 63y) = 12 + (-98)
-28x - 20y + 28x + 63y = -86
43y = -86
```
- When we add the two equations, the terms -28x and +28x cancel each other out, leaving us with an equation in terms of y only. This is the essence of the elimination method: simplifying the system to a single equation with a single variable. The resulting equation, 43y = -86, is much easier to solve than the original system. By eliminating x, we've effectively reduced the problem to finding the value of y. This step is crucial because it transforms a complex system into a straightforward equation, which can be solved using basic algebraic techniques. The cancellation of terms highlights the elegance and efficiency of the elimination method.
Solve for the remaining variable.
- Divide both sides by 43:
```
y = -86 / 43
y = -2
```
- By dividing both sides of the equation 43y = -86 by 43, we isolate y and find its value to be -2. This is a significant milestone in solving the system of equations. Knowing the value of y allows us to proceed to find the value of x. The simplicity of this step underscores the power of the elimination method in simplifying complex problems. Once we have solved for one variable, we can substitute its value back into one of the original equations to find the value of the other variable. This process of solving for one variable and then substituting its value is a common technique in solving systems of equations.
Substitute the value back into one of the original equations to solve for the other variable.
- Substitute y = -2 into the first original equation:
```
-7x - 5(-2) = 3
-7x + 10 = 3
-7x = 3 - 10
-7x = -7
x = -7 / -7
x = 1
```
- Substituting y = -2 into the first original equation allows us to solve for x. We replace y with -2 in the equation -7x - 5y = 3, which simplifies to -7x + 10 = 3. By subtracting 10 from both sides, we get -7x = -7. Finally, dividing both sides by -7, we find that x = 1. This completes the solution process, as we have now found the values of both x and y. The substitution step is a crucial part of solving systems of equations, as it allows us to use the value of one variable to find the value of the other. This process highlights the interconnectedness of the variables in the system.
State the solution.
- The solution to the system of equations is x = 1 and y = -2.

Verify the Solution

To ensure the accuracy of our solution, we should always verify it by substituting the values of x and y back into both original equations. This step helps to catch any errors made during the solving process. By plugging the values into both equations, we can confirm that they satisfy both conditions simultaneously. This verification process is a fundamental practice in mathematics, as it provides a high level of confidence in the correctness of the solution. It is particularly important in exams and assessments, where showing your verification steps can earn you extra points. Moreover, it is a great way to reinforce your understanding of the solution process and the concept of solving systems of equations.

Verification Steps

Substitute x = 1 and y = -2 into the first equation:
```
-7(1) - 5(-2) = 3
-7 + 10 = 3
3 = 3 (True)
```
Substitute x = 1 and y = -2 into the second equation:
```
4(1) + 9(-2) = -14
4 - 18 = -14
-14 = -14 (True)
```

Since the values satisfy both equations, our solution is correct.

Conclusion

We have successfully solved the given system of equations using the elimination method. The solution is x = 1 and y = -2. This method provides a systematic approach to solving linear systems and is a valuable tool in algebra. By mastering the elimination method, you gain a powerful technique for solving a wide range of mathematical problems. This skill is essential not only in algebra but also in various fields, including engineering, physics, and economics, where systems of equations frequently arise. The step-by-step process we followed, from manipulating the equations to eliminating variables and verifying the solution, provides a clear and concise framework for tackling similar problems. Remember, practice is key to mastering any mathematical technique, so work through various examples to solidify your understanding and build your confidence.