Solving Systems Of Equations Find The Solution

Jul 15, 2025 by ADMIN 47 views

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 containing 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. Finding these solutions is a fundamental concept in algebra and has numerous applications in various fields, including science, engineering, economics, and computer science. This guide will walk you through the process of solving systems of equations, focusing on the substitution method, and will provide a detailed solution to the specific system presented.

Understanding Systems of Equations

Before diving into the solution, let's clarify what a system of equations is and why it's important. A system of equations is a collection of two or more equations that share the same set of variables. For example:

y = -5x + 3
y = 1

This is a system of two equations with two variables, x and y. The solution to this system is a pair of values (x, y) that satisfies both equations. Graphically, each equation represents a line, and the solution is the point where these lines intersect. Understanding systems of equations is crucial because they model real-world scenarios where multiple conditions must be met simultaneously. For instance, in economics, you might use a system of equations to find the equilibrium point where supply and demand curves intersect. In engineering, you might use them to analyze the forces acting on a structure.

Methods for Solving Systems of Equations

There are several methods to solve systems of equations, each with its advantages and disadvantages. The most common methods include:

Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation.
Elimination (or Addition): This method involves adding or subtracting the equations to eliminate one variable.
Graphing: This method involves graphing the equations and finding the point(s) of intersection.
Matrix Methods: These methods use matrix algebra to solve systems of linear equations, especially useful for larger systems.

For this particular problem, the substitution method is the most straightforward approach due to the simplicity of the second equation (y = 1). Substitution is particularly effective when one of the equations is already solved for one of the variables or can be easily solved.

Solving the System by Substitution

The given system of equations is:

y = -5x + 3
y = 1

Our goal is to find the values of x and y that satisfy both equations. Since the second equation already gives us the value of y, we can use the substitution method.

Step 1: Substitute the value of y from the second equation into the first equation.

We know that y = 1. Substituting this into the first equation, we get:

1 = -5x + 3

Step 2: Solve the resulting equation for x.

Now we have a single equation with one variable, x. To solve for x, we need to isolate it on one side of the equation.

Subtract 3 from both sides:

1 - 3 = -5x + 3 - 3
-2 = -5x

Divide both sides by -5:

-2 / -5 = -5x / -5
2/5 = x

So, x = 2/5, which is equal to 0.4.

Step 3: Write the solution as an ordered pair (x, y).

We found that x = 0.4 and we were given that y = 1. Therefore, the solution to the system of equations is (0.4, 1).

Verifying the Solution

To ensure our solution is correct, we can substitute the values of x and y back into both original equations and check if they hold true.

Equation 1: y = -5x + 3

Substitute x = 0.4 and y = 1:

1 = -5(0.4) + 3
1 = -2 + 3
1 = 1

The equation holds true.

Equation 2: y = 1

This equation is already satisfied since we know y = 1.

Since the solution (0.4, 1) satisfies both equations, we have verified that it is the correct solution.

Analyzing the Options

Now let's look at the given options:

A. (0.4, 1) B. (0.8, 1) C. (1, 0.4) D. (1, 0.8)

Our solution is (0.4, 1), which corresponds to option A.

Why Other Options Are Incorrect

It's helpful to understand why the other options are incorrect. This reinforces the concept of what a solution to a system of equations truly means.

Option B (0.8, 1): If we substitute x = 0.8 and y = 1 into the first equation, we get:
```
1 = -5(0.8) + 3
1 = -4 + 3
1 = -1
```
This is not true, so (0.8, 1) is not a solution.
Option C (1, 0.4): If we substitute x = 1 and y = 0.4 into the first equation, we get:
```
0.  4 = -5(1) + 3
0.  4 = -5 + 3
0.  4 = -2
```
This is not true, so (1, 0.4) is not a solution.
Option D (1, 0.8): If we substitute x = 1 and y = 0.8 into the first equation, we get:
```
0.  8 = -5(1) + 3
0.  8 = -5 + 3
0.  8 = -2
```
This is not true, so (1, 0.8) is not a solution.

Only option A (0.4, 1) satisfies both equations in the system.

Conclusion

The solution to the system of equations:

y = -5x + 3
y = 1

is (0.4, 1). We arrived at this solution by using the substitution method, which is a powerful technique for solving systems of equations. By substituting the value of y from the second equation into the first equation, we were able to solve for x. Then, we verified our solution by plugging the values of x and y back into the original equations. Understanding how to solve systems of equations is a fundamental skill in mathematics with wide-ranging applications. Practice with different systems and methods will solidify your understanding and make you a more confident problem solver.

This comprehensive guide has walked you through the process of solving a system of equations using the substitution method. Remember, the key is to understand the underlying concepts and apply the appropriate method strategically. With practice, you'll become proficient at solving various types of systems of equations.