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Solve The Following System Of Linear Equations

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Nov 12, 2025 · 11 min read

Solve The Following System Of Linear Equations
Solve The Following System Of Linear Equations

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    Solving a system of linear equations is a fundamental task in mathematics, with applications spanning various fields such as engineering, economics, computer science, and physics. A system of linear equations consists of two or more linear equations involving the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. This comprehensive guide delves into several methods for solving such systems, from the basic to the more advanced, providing detailed explanations and examples to ensure a thorough understanding.

    Introduction to Systems of Linear Equations

    A linear equation is an equation in which the highest power of any variable is one. A system of linear equations, therefore, is a set of two or more linear equations that are considered together. For example:

    2x + y = 7
x - y = -1

    Here, we have a system of two linear equations with two variables, x and y. A solution to this system is a pair of values for x and y that makes both equations true.

    Types of Solutions

    When solving a system of linear equations, there are three possible outcomes:

    1. Unique Solution: The system has exactly one solution, meaning there is only one set of values for the variables that satisfies all equations.
    2. No Solution: The system is inconsistent, meaning there is no set of values for the variables that can satisfy all equations simultaneously.
    3. Infinite Solutions: The system is dependent, meaning there are infinitely many sets of values for the variables that satisfy all equations. This typically occurs when the equations are multiples of each other or represent the same line.

    Methods for Solving Systems of Linear Equations

    There are several methods to solve systems of linear equations, each with its advantages and suited for different types of systems. The most common methods include:

    1. Substitution Method
    2. Elimination Method
    3. Gaussian Elimination (Row Echelon Form)
    4. Matrix Methods (Inverse and Cramer's Rule)
    5. Graphical Method

    We will explore each of these methods in detail.

    1. Substitution Method

    The substitution method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.

    Steps for Substitution Method:

    1. Solve one equation for one variable: Choose one equation and solve it for one of the variables. Pick the equation and variable that are easiest to isolate.
    2. Substitute: Substitute the expression found in step 1 into the other equation.
    3. Solve for the remaining variable: Solve the new equation for the remaining variable.
    4. Back-substitute: Substitute the value found in step 3 back into the expression from step 1 to find the value of the other variable.
    5. Check the solution: Verify that the solution satisfies both original equations.

    Example:

    Solve the following system of equations using the substitution method:

    x + y = 5   (1)
2x - y = 1  (2)

    Solution:

    1. Solve equation (1) for x:

      x = 5 - y

    2. Substitute x in equation (2):

      2(5 - y) - y = 1

    3. Solve for y:

      10 - 2y - y = 1
10 - 3y = 1
-3y = -9
y = 3

    4. Back-substitute y = 3 into x = 5 - y:

      x = 5 - 3
x = 2

    5. Check the solution:

      • Equation (1): 2 + 3 = 5 (True)
      • Equation (2): 2(2) - 3 = 1 (True)

    Therefore, the solution to the system of equations is x = 2 and y = 3.

    2. Elimination Method

    The elimination method, also known as the addition method, involves adding or subtracting the equations in the system to eliminate one of the variables.

    Steps for Elimination Method:

    1. Align the equations: Write the equations so that like terms are in columns.
    2. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one of the variables are opposites.
    3. Add the equations: Add the equations together to eliminate one of the variables.
    4. Solve for the remaining variable: Solve the resulting equation for the remaining variable.
    5. Back-substitute: Substitute the value found in step 4 back into one of the original equations to find the value of the other variable.
    6. Check the solution: Verify that the solution satisfies both original equations.

    Example:

    Solve the following system of equations using the elimination method:

    2x + y = 7   (1)
x - y = -1  (2)

    Solution:

    1. Align the equations:

      2x + y = 7
x - y = -1

    2. Add the equations:

      (2x + y) + (x - y) = 7 + (-1)
3x = 6

    3. Solve for x:

      x = 2

    4. Back-substitute x = 2 into equation (1):

      2(2) + y = 7
4 + y = 7
y = 3

    5. Check the solution:

      • Equation (1): 2(2) + 3 = 7 (True)
      • Equation (2): 2 - 3 = -1 (True)

    Therefore, the solution to the system of equations is x = 2 and y = 3.

    3. Gaussian Elimination (Row Echelon Form)

    Gaussian elimination is a method for solving systems of linear equations by transforming the system into an equivalent system in row echelon form or reduced row echelon form. This method is particularly useful for larger systems of equations.

    Steps for Gaussian Elimination:

    1. Write the augmented matrix: Represent the system of equations as an augmented matrix.
    2. Transform to row echelon form: Use elementary row operations to transform the matrix into row echelon form. This means:
      • The first non-zero entry (leading entry) in each row is 1.
      • Each leading entry is in a column to the right of the leading entry in the row above it.
      • Rows with all zero entries are at the bottom of the matrix.
    3. Back-substitution: Use back-substitution to solve for the variables.

    Elementary Row Operations:

    1. Interchange two rows: Swap the positions of two rows.
    2. Multiply a row by a non-zero constant: Multiply all entries in a row by the same non-zero constant.
    3. Add a multiple of one row to another row: Add a multiple of one row to another row.

    Example:

    Solve the following system of equations using Gaussian elimination:

    x + y + z = 6    (1)
2x - y + z = 3   (2)
x + 2y - z = 2   (3)

    Solution:

    1. Write the augmented matrix:

      [ 1  1  1 | 6 ]
[ 2 -1  1 | 3 ]
[ 1  2 -1 | 2 ]

    2. Transform to row echelon form:

      • Step 1: Eliminate the x term in the second and third rows.

        • Replace Row 2 with Row 2 - 2 * Row 1:

          [ 1  1  1 | 6 ]
[ 0 -3 -1 | -9 ]
[ 1  2 -1 | 2 ]

        • Replace Row 3 with Row 3 - Row 1:

          [ 1  1  1 | 6 ]
[ 0 -3 -1 | -9 ]
[ 0  1 -2 | -4 ]

      • Step 2: Eliminate the y term in the third row.

        • Multiply Row 2 by -1/3 to make the leading entry 1:

          [ 1  1  1 | 6 ]
[ 0  1  1/3 | 3 ]
[ 0  1 -2 | -4 ]

        • Replace Row 3 with Row 3 - Row 2:

          [ 1  1  1 | 6 ]
[ 0  1  1/3 | 3 ]
[ 0  0 -7/3 | -7 ]

      • Step 3: Make the leading entry in the third row 1:

        • Multiply Row 3 by -3/7:

          [ 1  1  1 | 6 ]
[ 0  1  1/3 | 3 ]
[ 0  0  1 | 3 ]

      The matrix is now in row echelon form.

    3. Back-substitution:

      • From Row 3: z = 3
      • From Row 2: y + (1/3)z = 3 => y + (1/3)(3) = 3 => y = 2
      • From Row 1: x + y + z = 6 => x + 2 + 3 = 6 => x = 1

    Therefore, the solution to the system of equations is x = 1, y = 2, and z = 3.

    4. Matrix Methods (Inverse and Cramer's Rule)

    Matrix methods provide powerful tools for solving systems of linear equations, particularly when dealing with larger systems. Two common matrix methods are the inverse matrix method and Cramer's Rule.

    a. Inverse Matrix Method

    This method involves expressing the system of equations in matrix form and using the inverse of the coefficient matrix to solve for the variables.

    Steps for Inverse Matrix Method:
    1. Write the matrix equation: Express the system of equations as AX = B, where:
      • A is the coefficient matrix.
      • X is the column matrix of variables.
      • B is the column matrix of constants.
    2. Find the inverse of A: Calculate the inverse of the coefficient matrix A, denoted as A<sup>-1</sup>.
    3. Solve for X: Multiply both sides of the matrix equation by A<sup>-1</sup> to solve for X: X = A<sup>-1</sup>B.
    Example:

    Solve the following system of equations using the inverse matrix method:

    2x + y = 7
x - y = -1

    Solution:

    1. Write the matrix equation:

      A = [ 2  1 ]
    [ 1 -1 ]

X = [ x ]
    [ y ]

B = [ 7 ]
    [ -1 ]

AX = B  =>  [ 2  1 ] [ x ] = [ 7 ]
            [ 1 -1 ] [ y ] = [ -1 ]

    2. Find the inverse of A:

      • Calculate the determinant of A: det(A) = (2)(-1) - (1)(1) = -2 - 1 = -3
      • Find the adjoint of A: adj(A) = [ -1 -1 ] [ -1 2 ]
      • Calculate A<sup>-1</sup>: A<sup>-1</sup> = (1/det(A)) * adj(A) = (-1/3) [ -1 -1 ] = [ 1/3 1/3 ] [ -1 2 ] [ 1/3 -2/3 ]

    3. Solve for X:

      X = A^(-1)B = [ 1/3  1/3 ] [ 7 ] = [ (1/3)(7) + (1/3)(-1) ] = [ 6/3 ] = [ 2 ]
               [ 1/3 -2/3 ] [ -1 ]   [ (1/3)(7) + (-2/3)(-1) ]   [ 9/3 ]   [ 3 ]

[ x ] = [ 2 ]
[ y ] = [ 3 ]

    Therefore, the solution to the system of equations is x = 2 and y = 3.

    b. Cramer's Rule

    Cramer's Rule is a method for solving systems of linear equations using determinants. It provides a direct way to find the values of the variables without the need for back-substitution.

    Steps for Cramer's Rule:

    1. Write the system in matrix form: Express the system of equations in the form AX = B.

    2. Calculate the determinant of A: Find the determinant of the coefficient matrix A, denoted as det(A).

    3. Calculate the determinants for each variable: For each variable, replace the corresponding column in A with the column matrix B and calculate the determinant. Let det(A<sub>x</sub>) be the determinant when the x column is replaced by B, det(A<sub>y</sub>) be the determinant when the y column is replaced by B, and so on.

    4. Solve for the variables: Use the following formulas to find the values of the variables:

      x = det(A_x) / det(A)
y = det(A_y) / det(A)
z = det(A_z) / det(A)
    Example:

    Solve the following system of equations using Cramer's Rule:

    2x + y = 7
x - y = -1

    Solution:

    1. Write the system in matrix form:

      A = [ 2  1 ]
    [ 1 -1 ]

B = [ 7 ]
    [ -1 ]

    2. Calculate the determinant of A:

      • det(A) = (2)(-1) - (1)(1) = -2 - 1 = -3

    3. Calculate the determinants for each variable:

      • Replace the x column with B:

        A_x = [ 7  1 ]
      [ -1 -1 ]
det(A_x) = (7)(-1) - (1)(-1) = -7 + 1 = -6

      • Replace the y column with B:

        A_y = [ 2  7 ]
      [ 1 -1 ]
det(A_y) = (2)(-1) - (7)(1) = -2 - 7 = -9

    4. Solve for the variables:

      x = det(A_x) / det(A) = -6 / -3 = 2
y = det(A_y) / det(A) = -9 / -3 = 3

    Therefore, the solution to the system of equations is x = 2 and y = 3.

    5. Graphical Method

    The graphical method involves plotting the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system.

    Steps for Graphical Method:

    1. Rewrite equations in slope-intercept form: Rewrite each equation in the form y = mx + b, where m is the slope and b is the y-intercept.
    2. Plot the lines: Plot each equation on the same coordinate plane.
    3. Find the intersection point: Identify the point where the lines intersect. The coordinates of this point represent the solution to the system of equations.
    4. Verify the solution: Check that the solution satisfies both original equations.

    Example:

    Solve the following system of equations using the graphical method:

    x + y = 5
2x - y = 1

    Solution:

    1. Rewrite equations in slope-intercept form:

      • Equation (1): y = -x + 5
      • Equation (2): y = 2x - 1

    2. Plot the lines: Plot the lines y = -x + 5 and y = 2x - 1 on the same coordinate plane.

    3. Find the intersection point: The lines intersect at the point (2, 3).

    4. Verify the solution:

      • Equation (1): 2 + 3 = 5 (True)
      • Equation (2): 2(2) - 3 = 1 (True)

    Therefore, the solution to the system of equations is x = 2 and y = 3.

    Conclusion

    Solving systems of linear equations is a critical skill in mathematics and has numerous applications in various fields. This guide has provided a comprehensive overview of several methods for solving such systems, including substitution, elimination, Gaussian elimination, matrix methods (inverse and Cramer's Rule), and the graphical method. Each method has its strengths and is suited for different types of systems. By understanding these methods, one can effectively solve a wide range of linear equation systems.

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