Determining Solutions Of Linear Systems A Step-by-Step Guide
In the realm of mathematics, linear systems of equations play a pivotal role. They serve as the foundation for modeling and solving a myriad of real-world problems across various disciplines, including engineering, economics, and computer science. One of the fundamental questions that arises when dealing with linear systems is the nature of their solutions – do they have a unique solution, infinitely many solutions, or no solution at all? In this comprehensive guide, we will delve into the intricacies of determining the number of solutions a linear system possesses, using the given system as a case study:
y = 2x - 5
-8x - 4y = -20
Our exploration will involve a step-by-step analysis, leveraging algebraic techniques and graphical interpretations to unravel the solution landscape of this system. By the end of this guide, you will not only be able to confidently determine the number of solutions for this particular system but also gain a solid understanding of the underlying principles applicable to a broader range of linear systems.
Understanding Linear Systems and Their Solutions
Before we dive into the specifics of the given system, let's establish a firm grasp of the fundamental concepts surrounding linear systems and their solutions. A linear system, at its core, is a collection of two or more linear equations involving the same set of variables. A linear equation, in turn, is an equation in which the highest power of any variable is one. Graphically, linear equations represent straight lines, and the solutions to a linear system correspond to the points where these lines intersect.
When analyzing a linear system, there are three possible scenarios regarding the number of solutions:
- Unique Solution: The system has exactly one solution, which corresponds to the single point where the lines intersect. This scenario is characterized by the lines having different slopes.
- Infinitely Many Solutions: The system has an infinite number of solutions, which occurs when the equations represent the same line. In this case, the lines coincide, and every point on the line is a solution to the system.
- No Solution: The system has no solution, which arises when the lines are parallel and do not intersect. Parallel lines have the same slope but different y-intercepts.
To determine the number of solutions for a given linear system, we can employ various techniques, including substitution, elimination, and graphical methods. In the following sections, we will apply these methods to the system at hand to unveil its solution characteristics.
Solving the Linear System: A Step-by-Step Approach
Now, let's embark on the journey of solving the given linear system:
y = 2x - 5
-8x - 4y = -20
We will employ the substitution method to find the solutions for this system. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process eliminates one variable, allowing us to solve for the remaining variable.
Step 1: Solve one equation for one variable.
The first equation, y = 2x - 5, is already solved for y. This makes it an ideal starting point for the substitution method.
Step 2: Substitute the expression into the other equation.
Substitute the expression 2x - 5 for y in the second equation:
-8x - 4(2x - 5) = -20
Step 3: Simplify and solve for the remaining variable.
Distribute the -4 and simplify the equation:
-8x - 8x + 20 = -20
-16x + 20 = -20
Subtract 20 from both sides:
-16x = -40
Divide both sides by -16:
x = 2.5
Step 4: Substitute the value back into either equation to find the other variable.
Substitute x = 2.5 into the first equation, y = 2x - 5:
y = 2(2.5) - 5
y = 5 - 5
y = 0
Therefore, the solution to the linear system is x = 2.5 and y = 0.
Graphical Interpretation: Visualizing the Solution
To gain a deeper understanding of the solution, let's visualize the system graphically. Each equation represents a straight line, and the solution corresponds to the point where these lines intersect. To plot the lines, we can find two points on each line and connect them.
For the equation y = 2x - 5:
- When
x = 0,y = -5. So, one point is (0, -5). - When
x = 2.5,y = 0. So, another point is (2.5, 0).
For the equation -8x - 4y = -20, we can rewrite it in slope-intercept form (y = mx + b) to make it easier to plot:
-4y = 8x - 20
y = -2x + 5
- When
x = 0,y = 5. So, one point is (0, 5). - When
x = 2.5,y = 0. So, another point is (2.5, 0).
Plotting these lines, we observe that they intersect at the point (2.5, 0), which confirms our algebraic solution. The graphical representation provides a visual affirmation that the system has a unique solution.
Alternative Approach: Transforming the Equations
Let's explore an alternative approach to solving the system by manipulating the equations to reveal their relationship more directly.
Consider the second equation:
-8x - 4y = -20
Divide both sides of the equation by -4:
2x + y = 5
Now, solve for y:
y = -2x + 5
Comparing this transformed equation with the first equation, y = 2x - 5, we notice that they have different slopes (2 and -2). This difference in slopes indicates that the lines are not parallel and will intersect at a single point. Therefore, the system has a unique solution.
However, there seems to be a mistake in the simplification. Let's correct it. Starting with the second equation:
-8x - 4y = -20
Divide both sides by -4:
2x + y = 5
Now, solve for y:
y = -2x + 5
This is where the key observation comes in. The initial assessment that there was a single solution was based on an error. If we carefully analyze the two equations, y = 2x - 5 and 2x + y = 5 (rewritten as y = -2x + 5), we see that they represent two different lines because they have different slopes. The slopes are 2 and -2 respectively. This means they will intersect at one point, confirming a single solution.
Determining the Number of Solutions: Key Indicators
Based on our analysis, we can identify key indicators that help determine the number of solutions a linear system possesses:
- Different Slopes: If the equations in the system have different slopes, the lines will intersect at a single point, indicating a unique solution.
- Same Slope and Different Y-intercepts: If the equations have the same slope but different y-intercepts, the lines are parallel and will never intersect, resulting in no solution.
- Same Slope and Same Y-intercept: If the equations have the same slope and the same y-intercept, they represent the same line, leading to infinitely many solutions.
In the given system, the equations have different slopes (2 and -2), confirming our finding that the system has a unique solution.
Conclusion: One Unique Solution
In this comprehensive guide, we embarked on a journey to determine the number of solutions for the linear system:
y = 2x - 5
-8x - 4y = -20
Through the application of algebraic techniques, graphical interpretation, and alternative equation transformations, we definitively established that this system has a unique solution. The solution, x = 2.5 and y = 0, represents the single point where the lines corresponding to the equations intersect.
This exploration not only provided a solution to the specific problem but also equipped you with a solid understanding of the principles governing the solution landscape of linear systems. By recognizing the key indicators, such as the slopes and y-intercepts of the equations, you can confidently determine the number of solutions for a wide range of linear systems encountered in mathematics and its diverse applications.
