Determining The Number Of Solutions For A System Of Equations

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How many solutions are there for the following system of equations? 3x + y = 18 and 3x + y = 16

In mathematics, solving a system of equations is a fundamental concept. It involves finding the values of the variables that satisfy all the equations in the system simultaneously. The number of solutions can vary, and understanding the conditions that lead to different outcomes is crucial. This article will delve into the process of determining the number of solutions for a given system of linear equations, using the example provided.

Understanding Systems of Linear Equations

Before we tackle the specific problem, let's establish a solid understanding of systems of linear equations. A system of linear equations is a set of two or more linear equations containing the same variables. A linear equation is one in which the highest power of any variable is 1. Graphically, a linear equation in two variables represents a straight line. The solution to a system of linear equations is the set of values for the variables that make all the equations true. Geometrically, this corresponds to the point(s) where the lines representing the equations intersect.

When dealing with two linear equations in two variables, there are three possible scenarios regarding the number of solutions:

  1. One Solution: The lines intersect at a single point. This means there is a unique set of values for the variables that satisfies both equations.
  2. No Solution: The lines are parallel and do not intersect. In this case, there are no values for the variables that can satisfy both equations simultaneously.
  3. Infinitely Many Solutions: The lines are coincident, meaning they overlap completely. Any point on the line represents a solution to both equations, leading to an infinite number of solutions.

To determine which scenario applies to a given system of equations, we can use various methods, such as substitution, elimination, or graphical analysis. We will explore these methods in the context of the given problem.

Analyzing the Given System of Equations

The system of equations we need to analyze is:

3x + y = 18
3x + y = 16

This system consists of two linear equations in two variables, x and y. Our goal is to determine whether this system has none, one, two, or infinitely many solutions.

Method 1: Elimination

The elimination method involves manipulating the equations in a way that allows us to eliminate one of the variables. In this case, we can easily eliminate x by subtracting one equation from the other. Let's subtract the second equation from the first:

(3x + y) - (3x + y) = 18 - 16

Simplifying the equation, we get:

0 = 2

This statement is clearly false. The equation 0 = 2 is a contradiction, indicating that there is no solution to the system of equations. This is because the two equations represent parallel lines that never intersect. Parallel lines have the same slope but different y-intercepts. In this case, both lines have a slope of -3, but the first line has a y-intercept of 18 and the second has a y-intercept of 16.

Method 2: Graphical Analysis

Another way to visualize the system is to graph the two equations. To graph a linear equation, we can rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

The first equation, 3x + y = 18, can be rewritten as:

y = -3x + 18

The second equation, 3x + y = 16, can be rewritten as:

y = -3x + 16

As we can see, both equations have the same slope (-3) but different y-intercepts (18 and 16). This confirms that the lines are parallel. When we graph these lines, we would observe that they never intersect, further demonstrating that there is no solution to the system.

Method 3: Comparing Coefficients

A more direct way to determine the number of solutions is by comparing the coefficients of the variables and the constant terms in the equations. Consider the general form of a system of two linear equations in two variables:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

The number of solutions can be determined by the following conditions:

  • One Solution: If a₁/a₂ ≠ b₁/b₂
  • No Solution: If a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  • Infinitely Many Solutions: If a₁/a₂ = b₁/b₂ = c₁/c₂

In our system:

3x + y = 18
3x + y = 16

We have a₁ = 3, b₁ = 1, c₁ = 18, a₂ = 3, b₂ = 1, and c₂ = 16. Let's calculate the ratios:

a₁/a₂ = 3/3 = 1
b₁/b₂ = 1/1 = 1
c₁/c₂ = 18/16 = 9/8

Since a₁/a₂ = b₁/b₂ but a₁/a₂ ≠ c₁/c₂, the system has no solution.

Conclusion

Through multiple methods of analysis, we have determined that the system of equations:

3x + y = 18
3x + y = 16

has no solution. This is because the equations represent parallel lines that never intersect. Understanding the conditions that lead to different solution scenarios is crucial for solving systems of linear equations effectively. By using techniques such as elimination, graphical analysis, and comparing coefficients, we can confidently determine the number of solutions for a given system.

Therefore, the correct answer is A. none.

This exercise demonstrates the importance of careful analysis when dealing with systems of equations. While the equations may appear similar at first glance, a closer examination reveals the inconsistency that leads to the absence of a solution. Mastering these techniques will empower you to solve a wide range of mathematical problems involving linear equations and beyond.

Further Exploration

To further enhance your understanding of systems of equations, consider exploring the following topics:

  • Solving Systems with Three Variables: Extend your knowledge to systems involving three or more variables and equations.
  • Matrix Methods: Learn how to use matrices to represent and solve systems of equations, a powerful technique in linear algebra.
  • Applications of Systems of Equations: Discover real-world applications of systems of equations in fields such as physics, engineering, and economics.

By delving deeper into these areas, you can develop a more comprehensive understanding of systems of equations and their significance in various disciplines. Remember, practice is key to mastering any mathematical concept. Work through numerous examples and challenges to solidify your skills and build confidence in your problem-solving abilities.