Solving Linear Equations The System X-3y=-2 And X+3y=16

by ADMIN 56 views

In the realm of mathematics, systems of linear equations are fundamental concepts with wide-ranging applications in various fields, including engineering, economics, and computer science. These systems involve two or more linear equations with the same set of variables, and the solution to such a system represents the values of the variables that satisfy all equations simultaneously. This article delves into the methods for solving systems of linear equations, providing a step-by-step guide to finding solutions and illustrating the concepts with a practical example.

Understanding Systems of Linear Equations

Before diving into the solution methods, it's crucial to grasp the essence of a system of linear equations. A linear equation is an algebraic expression in which the highest power of the variables is one. A system of linear equations, therefore, comprises two or more linear equations involving the same variables. The goal is to find the values of these variables that make all equations in the system true.

Geometrically, each linear equation in a two-variable system represents a straight line on a coordinate plane. The solution to the system corresponds to the point(s) where these lines intersect. If the lines intersect at a single point, the system has a unique solution. If the lines coincide, the system has infinitely many solutions. If the lines are parallel and do not intersect, the system has no solution.

Systems of linear equations can be classified based on the number of solutions they possess:

  • Consistent Systems: Systems that have at least one solution.
  • Inconsistent Systems: Systems that have no solutions.
  • Independent Systems: Systems with a unique solution.
  • Dependent Systems: Systems with infinitely many solutions.

Methods for Solving Systems of Linear Equations

Several methods are available for solving systems of linear equations, each with its strengths and suitability for different types of systems. We will explore the following common methods:

  1. Graphical Method: This method involves plotting the equations on a coordinate plane and visually identifying the point(s) of intersection, which represent the solution(s) to the system. It is particularly useful for systems with two variables and provides a visual understanding of the solution.
  2. Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable.
  3. Elimination Method (also known as the Addition Method): This method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, this variable is eliminated, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable.
  4. Matrix Methods: These methods, such as Gaussian elimination and matrix inversion, are particularly useful for solving systems with three or more variables. They involve representing the system of equations in matrix form and applying matrix operations to solve for the variables.

1. Graphical Method: Visualizing Solutions

The graphical method provides a visual representation of the solutions to a system of linear equations. Each equation in the system represents a line on a coordinate plane, and the point(s) where these lines intersect correspond to the solution(s) of the system. This method is particularly useful for systems with two variables, as it allows for a clear visualization of the relationship between the equations and their solutions.

To implement the graphical method, follow these steps:

  1. Rewrite each equation in slope-intercept form (y = mx + b): This form makes it easy to identify the slope (m) and y-intercept (b) of each line.
  2. Plot each line on the coordinate plane: Use the slope and y-intercept to plot at least two points for each line, and then draw a line through these points.
  3. Identify the point(s) of intersection: The point(s) where the lines intersect represent the solution(s) to the system. The coordinates of the intersection point(s) are the values of the variables that satisfy both equations.
  • If the lines intersect at a single point, the system has a unique solution.
  • If the lines coincide (are the same line), the system has infinitely many solutions.
  • If the lines are parallel and do not intersect, the system has no solution.

The graphical method is a valuable tool for understanding the nature of solutions to systems of linear equations. It provides a visual representation of the relationships between the equations and their solutions, making it easier to grasp the concept of simultaneous solutions.

2. Substitution Method: Solving for One Variable at a Time

The substitution method is an algebraic technique for solving systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This process eliminates one variable, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is particularly effective when one of the equations can be easily solved for one variable in terms of the other.

The steps for the substitution method are as follows:

  1. Solve one equation for one variable: Choose one of the equations and solve it for one variable in terms of the other variable. This means isolating one variable on one side of the equation.
  2. Substitute the expression into the other equation: Substitute the expression you obtained in step 1 into the other equation. This will result in an equation with only one variable.
  3. Solve the resulting equation: Solve the equation from step 2 for the remaining variable.
  4. Substitute the value back into either original equation: Substitute the value you found in step 3 back into either of the original equations to solve for the other variable.
  5. Check your solution: Substitute the values you found for both variables into both original equations to verify that they satisfy both equations.

The substitution method is a powerful technique for solving systems of linear equations. It is particularly useful when one of the equations can be easily solved for one variable, making the substitution process straightforward. By systematically eliminating variables, this method leads to a solution for the system.

3. Elimination Method: Adding Equations to Eliminate Variables

The elimination method, also known as the addition method, is another algebraic technique for solving systems of linear equations. This method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, this variable is eliminated, allowing you to solve for the remaining variable. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is particularly effective when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant.

The steps for the elimination method are as follows:

  1. Multiply one or both equations by a constant (if necessary): The goal is to make the coefficients of one variable opposites. This may require multiplying one or both equations by a constant so that the coefficients of one variable are the same magnitude but opposite signs.
  2. Add the equations together: Add the two equations together. This will eliminate one variable, leaving you with an equation with only one variable.
  3. Solve the resulting equation: Solve the equation from step 2 for the remaining variable.
  4. Substitute the value back into either original equation: Substitute the value you found in step 3 back into either of the original equations to solve for the other variable.
  5. Check your solution: Substitute the values you found for both variables into both original equations to verify that they satisfy both equations.

The elimination method is a valuable tool for solving systems of linear equations. It is particularly useful when the coefficients of one variable are opposites or can be easily made opposites. By strategically eliminating variables, this method leads to a solution for the system.

Solving the Given System of Linear Equations

Now, let's apply these methods to solve the given system of linear equations:

 x - 3y = -2
 x + 3y = 16

Method 1: Elimination Method

Notice that the coefficients of y are already opposites (-3 and 3). So, we can directly add the two equations:

 (x - 3y) + (x + 3y) = -2 + 16
 2x = 14
 x = 7

Now, substitute x = 7 into either equation. Let's use the first equation:

 7 - 3y = -2
 -3y = -9
 y = 3

Therefore, the solution is x = 7 and y = 3.

Method 2: Substitution Method

Solve the first equation for x:

 x = 3y - 2

Substitute this expression for x into the second equation:

 (3y - 2) + 3y = 16
 6y - 2 = 16
 6y = 18
 y = 3

Now, substitute y = 3 back into the expression for x:

 x = 3(3) - 2
 x = 9 - 2
 x = 7

Again, the solution is x = 7 and y = 3.

Conclusion

Systems of linear equations are a fundamental concept in mathematics with numerous applications. This article has explored various methods for solving these systems, including the graphical method, substitution method, and elimination method. By understanding these methods and their underlying principles, you can effectively solve systems of linear equations and apply them to real-world problems. The example provided demonstrates the application of both the elimination and substitution methods to arrive at the same solution, highlighting the versatility of these techniques. Whether you are a student learning algebra or a professional applying mathematical concepts, mastering the solution of systems of linear equations is an invaluable skill. Remember to always check your solutions by substituting them back into the original equations to ensure accuracy.