How To Determine The Number Of Solutions In A System Of Equations

Jun 16, 2025 by ADMIN 66 views

How to Determine the Number of Solutions in a System of Equations

In the realm of mathematics, particularly when dealing with systems of equations, a fundamental question often arises: How many solutions does this system have? This question is not merely an academic exercise; it has profound implications in various fields, from engineering and economics to computer science and data analysis. Understanding the nature and number of solutions in a system of equations allows us to model real-world problems, make predictions, and optimize processes. In this comprehensive guide, we will delve deep into the methods and techniques used to determine the number of solutions in a system of equations, focusing on the example you provided and expanding to broader concepts.

Understanding Systems of Equations

Before we dive into the specifics of the given system, let's establish a clear understanding of what a system of equations entails. A system of equations is a set of two or more equations that share the same variables. The solutions to a system of equations are the values of the variables that satisfy all equations simultaneously. Geometrically, each equation in a two-variable system represents a line on a coordinate plane, and the solutions correspond to the points where these lines intersect. The number of solutions can be zero, one, or infinitely many, depending on the relationship between the lines.

Types of Solutions

Unique Solution: The system has exactly one solution, meaning the lines intersect at a single point. This occurs when the lines have different slopes.
No Solution: The system has no solution, meaning the lines are parallel and never intersect. This happens when the lines have the same slope but different y-intercepts.
Infinitely Many Solutions: The system has infinitely many solutions, meaning the lines are coincident (they are the same line). This occurs when the lines have the same slope and the same y-intercept.

Methods for Solving Systems of Equations

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

Substitution Method: Solving one equation for one variable and substituting that expression into the other equation.
Elimination Method: Adding or subtracting multiples of the equations to eliminate one variable.
Graphical Method: Plotting the equations on a coordinate plane and finding the points of intersection.
Matrix Methods: Using matrices and matrix operations to solve systems, especially useful for larger systems.

Analyzing the Given System

Now, let's focus on the specific system of equations you provided:

\begin{cases}
x + 5y = 45 \\
2x + y = 7
\end{cases}

To determine the number of solutions, we can employ several techniques. We will demonstrate the substitution and elimination methods to illustrate the process.

Method 1: Substitution

Solve one equation for one variable: Let's solve the second equation for y:
```
y = 7 - 2x
```
Substitute the expression for y into the first equation:
```
x + 5(7 - 2x) = 45
```
Simplify and solve for x:
```
x + 35 - 10x = 45
-9x = 10
x = -10/9
```

Substitute the value of x back into the expression for y:

y = 7 - 2(-10/9)
y = 7 + 20/9
y = (63 + 20)/9
y = 83/9

Therefore, we have found a unique solution: x = -10/9 and y = 83/9.

Method 2: Elimination

Multiply one or both equations by a constant so that the coefficients of one variable are opposites. Let's multiply the first equation by -2:
```
-2(x + 5y) = -2(45)
-2x - 10y = -90
```
Add the modified first equation to the second equation:
```
(-2x - 10y) + (2x + y) = -90 + 7
-9y = -83
```
Solve for y:
```
y = 83/9
```
Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
```
x + 5(83/9) = 45
x + 415/9 = 45
x = 45 - 415/9
x = (405 - 415)/9
x = -10/9
```

Again, we find the same unique solution: x = -10/9 and y = 83/9.

Conclusion for the Given System

Both the substitution and elimination methods have led us to the same conclusion: the system of equations has a unique solution. This means the two lines represented by the equations intersect at a single point on the coordinate plane. The solution is the ordered pair (-10/9, 83/9). This point satisfies both equations simultaneously, confirming that it is indeed the unique solution to the system.

Determining the Number of Solutions Without Solving

While solving the system provides the exact solution, sometimes we only need to know the number of solutions without finding them. We can determine this by examining the coefficients of the variables in the equations.

Slope-Intercept Form

One way to analyze the system is to rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Comparing the slopes and y-intercepts of the lines can quickly reveal the number of solutions.

Let's rewrite our equations:

x + 5y = 45
```
5y = -x + 45
y = (-1/5)x + 9
```
2x + y = 7
```
y = -2x + 7
```

Now, we can see that the first line has a slope of -1/5 and a y-intercept of 9, while the second line has a slope of -2 and a y-intercept of 7. Since the slopes are different, the lines intersect at one point, confirming that the system has a unique solution.

Comparing Coefficients

Another method involves comparing the ratios of the coefficients of the variables. For a system of two linear equations:

\begin{cases}
a_1x + b_1y = c_1 \\
a_2x + b_2y = c_2
\end{cases}

If a_1/a_2 ≠ b_1/b_2, the system has a unique solution.
If a_1/a_2 = b_1/b_2 ≠ c_1/c_2, the system has no solution (parallel lines).
If a_1/a_2 = b_1/b_2 = c_1/c_2, the system has infinitely many solutions (coincident lines).

Applying this to our system:

\begin{cases}
1x + 5y = 45 \\
2x + 1y = 7
\end{cases}

We have:

a_1 = 1, b_1 = 5, c_1 = 45
a_2 = 2, b_2 = 1, c_2 = 7

Let's compare the ratios:

a_1/a_2 = 1/2
b_1/b_2 = 5/1 = 5

Since 1/2 ≠ 5, the system has a unique solution, which aligns with our previous findings.

Generalizing to Larger Systems

The principles we've discussed extend to larger systems with more equations and variables. For instance, a system of three linear equations in three variables represents three planes in three-dimensional space. The solutions correspond to the points where the planes intersect. The system can have a unique solution (the planes intersect at a single point), no solution (the planes do not have a common intersection), or infinitely many solutions (the planes intersect along a line or are coincident).

Matrix Representation

For larger systems, matrix methods become particularly useful. A system of linear equations can be represented in matrix form as Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix. The determinant of the coefficient matrix A plays a crucial role in determining the number of solutions.

If the determinant of A is non-zero, the system has a unique solution.
If the determinant of A is zero, the system may have either no solution or infinitely many solutions, and further analysis is needed.

Row Echelon Form

Another powerful technique is to use Gaussian elimination or Gauss-Jordan elimination to transform the augmented matrix [A|b] into row echelon form or reduced row echelon form. The resulting form reveals the rank of the matrices, which helps determine the number of solutions. In summary:

Unique solution occurs when the number of pivots in row echelon form is equal to the number of variables.
Infinitely many solutions occur when the number of pivots is less than the number of variables, and the system is consistent.
No solution occurs when the number of pivots indicates inconsistency, such as a row in row echelon form where all coefficients are 0 but the constant term is not 0.

Real-World Applications

The ability to determine the number of solutions in a system of equations is not just a theoretical concept; it has numerous real-world applications. Here are a few examples:

Engineering: In structural engineering, systems of equations are used to analyze the forces and stresses in buildings and bridges. Determining whether a system has a unique solution ensures that the structure is stable and can withstand external loads.
Economics: Economic models often involve systems of equations to represent supply and demand, market equilibrium, and other economic relationships. The number of solutions can indicate whether the model has a stable equilibrium or whether there are multiple possible outcomes.
Computer Science: Systems of equations are used in computer graphics, image processing, and optimization algorithms. For instance, linear systems can be used to solve for the parameters of a transformation in computer graphics or to optimize the performance of a machine learning model.
Data Analysis: In data analysis, systems of equations can arise in regression analysis, curve fitting, and other statistical techniques. The number of solutions can provide insights into the reliability and accuracy of the models.

Conclusion

In conclusion, understanding how to determine the number of solutions in a system of equations is a fundamental skill in mathematics and its applications. For the system you provided:

\begin{cases}
x + 5y = 45 \\
2x + y = 7
\end{cases}

We have demonstrated using both substitution and elimination methods that the system has a unique solution. Furthermore, we have shown how to determine the number of solutions by analyzing the slopes and y-intercepts of the equations and by comparing the ratios of the coefficients.

This knowledge extends to larger systems, where matrix methods and row echelon forms become essential tools. The ability to analyze systems of equations has practical implications in various fields, making it a crucial skill for anyone working with mathematical models and problem-solving.

By mastering these techniques, you can confidently tackle systems of equations and unlock their potential in modeling and solving real-world problems. The process of understanding the nature and number of solutions is not just a mathematical exercise; it is a pathway to deeper insights and effective decision-making in a complex world.