Determining Solutions Of Linear System Y=2x-5, -8x-4y=-20

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Solving linear systems is a fundamental concept in mathematics, and understanding the number of solutions a system possesses is crucial. This article delves into the process of determining the number of solutions for a given linear system, using the example:

 y = 2x - 5
 -8x - 4y = -20

We will explore various methods to analyze this system and arrive at the correct answer, which in this case, is that the system has an infinite number of solutions. Let's embark on this mathematical journey!

Understanding Linear Systems and Their Solutions

In the realm of mathematics, a linear system, also known as a system of linear equations, comprises two or more linear equations involving the same set of variables. A solution to a linear system is a set of values for the variables that satisfies all equations simultaneously. Graphically, each linear equation represents a line, and the solution to the system corresponds to the point(s) where these lines intersect.

Linear systems can have one solution, no solution, or an infinite number of solutions. Let's break down each scenario:

  • One Solution: The lines intersect at exactly one point. This indicates that there is a unique set of values for the variables that satisfies both equations.
  • No Solution: The lines are parallel and never intersect. This means there is no set of values that can satisfy both equations simultaneously.
  • Infinite Number of Solutions: The lines are coincident, meaning they overlap completely. In this case, every point on the line represents a solution, resulting in an infinite number of solutions.

To determine the number of solutions for a given linear system, we can employ several methods, including substitution, elimination, and graphical analysis. In the following sections, we will apply these techniques to the given system and unravel its solution set.

Solving the Linear System: Substitution Method

The substitution method is a powerful algebraic technique for solving systems of equations. It involves solving one equation for one variable and substituting that expression into the other equation. This process eliminates one variable, allowing us to solve for the remaining variable. Let's apply this method to our linear system:

 y = 2x - 5
 -8x - 4y = -20

We observe that the first equation is already solved for y. So, we can substitute the expression 2x - 5 for y in the second equation:

 -8x - 4(2x - 5) = -20

Now, we simplify and solve for x:

 -8x - 8x + 20 = -20
 -16x = -40
 x = 2.5

We have found the value of x. Now, we substitute this value back into either of the original equations to solve for y. Let's use the first equation:

 y = 2(2.5) - 5
 y = 5 - 5
 y = 0

Thus, using the substitution method, we arrive at a potential solution: (2.5, 0). However, before we declare this as the unique solution, we need to verify if it satisfies both equations in the system.

Substituting x = 2.5 and y = 0 into the first equation:

 0 = 2(2.5) - 5
 0 = 5 - 5
 0 = 0 (True)

Substituting x = 2.5 and y = 0 into the second equation:

 -8(2.5) - 4(0) = -20
 -20 - 0 = -20
 -20 = -20 (True)

The solution (2.5, 0) satisfies both equations. However, this does not automatically mean it is the only solution. To definitively determine the number of solutions, we need to explore other methods or further analyze the system's properties.

Solving the Linear System: Elimination Method

The elimination method provides another powerful approach to solving systems of equations. This method involves manipulating the equations in the system to eliminate one variable, making it possible to solve for the remaining variable. Let's apply the elimination method to our system:

 y = 2x - 5
 -8x - 4y = -20

To effectively use the elimination method, we need to align the variables in both equations. Let's rewrite the first equation to have x and y terms on the same side:

 -2x + y = -5
 -8x - 4y = -20

Now, our goal is to eliminate either x or y. Observe that if we multiply the first equation by 4, the coefficient of y will become 4, which is the opposite of the coefficient of y in the second equation. This will allow us to eliminate y when we add the equations together.

Multiply the first equation by 4:

 4(-2x + y) = 4(-5)
 -8x + 4y = -20

Now we have the modified system:

 -8x + 4y = -20
 -8x - 4y = -20

Adding the two equations:

 (-8x + 4y) + (-8x - 4y) = -20 + (-20)
 -16x = -40

Solving for x:

 x = 2.5

Now substitute x = 2.5 into either of the original equations to find y. Let's use the original first equation:

 y = 2(2.5) - 5
 y = 5 - 5
 y = 0

Again, we arrive at the solution (2.5, 0). However, a closer look at the elimination process reveals something significant. When we added the modified equations, we obtained -16x = -40. If we divide both sides of the equation -8x + 4y = -20 by -4, we get:

 2x - y = 5

Rearranging this, we get:

 y = 2x - 5

This is identical to the first equation in the original system! This crucial observation indicates that the two equations in the system are essentially the same line. This means that any point that satisfies one equation will also satisfy the other. Therefore, there are infinitely many solutions.

Graphical Interpretation: Visualizing the Solution

Graphing the equations provides a visual confirmation of our findings. Each linear equation represents a line on the coordinate plane. The solution to the system is the point(s) where the lines intersect.

Let's graph the two equations:

  • y = 2x - 5
  • -8x - 4y = -20

To graph the second equation, we can rewrite it in slope-intercept form (y = mx + b):

 -4y = 8x - 20
 y = -2x + 5

Now we have the two equations in slope-intercept form:

  • y = 2x - 5
  • y = -2x + 5

Upon graphing these two lines, we would observe that they are the same line. This visually confirms that the system has an infinite number of solutions. Every point on the line represents a solution to the system.

Determining the Number of Solutions: A Summary

To determine the number of solutions a linear system has, we can use the following methods:

  1. Substitution Method: Solve one equation for one variable and substitute the expression into the other equation. If the resulting equation is always true (e.g., 0 = 0), the system has infinite solutions. If the resulting equation is a contradiction (e.g., 0 = 1), the system has no solution. If we find unique values for the variables, we have one solution, but we must verify that this solution satisfies both original equations.
  2. Elimination Method: Manipulate the equations to eliminate one variable. If we obtain an equation that is always true (e.g., 0 = 0), the system has infinite solutions. If we obtain a contradiction (e.g., 0 = 1), the system has no solution. If we find unique values for the variables, we have one solution, but we must verify that this solution satisfies both original equations.
  3. Graphical Analysis: Graph the equations. If the lines intersect at one point, there is one solution. If the lines are parallel, there is no solution. If the lines are coincident (overlap), there are infinite solutions.

In the given system:

 y = 2x - 5
 -8x - 4y = -20

We found that the two equations represent the same line. Therefore, the system has an infinite number of solutions.

Conclusion: Infinite Solutions Unveiled

Through the application of substitution, elimination, and graphical analysis, we have definitively determined that the linear system:

 y = 2x - 5
 -8x - 4y = -20

possesses an infinite number of solutions. This occurs because the two equations are dependent and represent the same line. Understanding how to analyze linear systems and determine their solution sets is a crucial skill in mathematics and has wide-ranging applications in various fields.

By mastering these techniques, you can confidently tackle linear systems and unravel their solutions, whether they be unique, nonexistent, or infinite. Remember to always verify your solutions and consider the graphical interpretation to solidify your understanding.