Systems of Equations: Three Equations and Three Unknowns

Systems of Equations: Three Equations and Three Unknowns

A system of three equations and three unknowns involves three linear equations with three variables. The general form is:

Where:

• $x$, $y$, and $z$ are the unknown variables.
• $a_1, a_2, a_3$, etc., are constants, as are $b_1, b_2, b_3$, and $c_1, c_2, c_3$.
• $d_1, d_2, d_3$ are the constants on the right-hand side of the equations.

The goal is to find the values of $x$, $y$, and $z$ that satisfy all three equations simultaneously.

Methods for Solving Systems of Equations with Three Unknowns

There are several methods to solve systems of three equations with three unknowns. The most common methods are:

1. Substitution Method
2. Elimination Method
3. Matrix Method (Gaussian Elimination)

Each method can be used to reduce the system to simpler equations and ultimately solve for the unknowns.

1. Substitution Method

The substitution method involves solving one of the equations for one variable and substituting that expression into the other two equations. This reduces the system to a system of two equations in two unknowns, which can then be solved.

Steps:

1. Solve one equation for one variable in terms of the others (e.g., solve for $x$, $y$, or $z$).
2. Substitute the expression into the other two equations, eliminating the variable.
3. Solve the resulting system of two equations in two unknowns.
4. Once you find the value of one variable, substitute it back into one of the original equations to find the other variables.

Example:

Solve the system:

• Step 1: Solve the first equation for $x$:

  $$x = 6 - y - z$$

• Step 2: Substitute $x = 6 - y - z$ into the second and third equations:

  • Second equation:

    $$2(6 - y - z) - y + 3z = 14 \quad \Rightarrow \quad 12 - 2y - 2z - y + 3z = 14 \quad \Rightarrow \quad 12 - 3y + z = 14$$

    Simplify:

    $$-3y + z = 2 \quad \text{(Equation 4)}$$

  • Third equation:

    $$3(6 - y - z) + 2y - z = 7 \quad \Rightarrow \quad 18 - 3y - 3z + 2y - z = 7 \quad \Rightarrow \quad 18 - y - 4z = 7$$

    Simplify:

    $$-y - 4z = -11 \quad \Rightarrow \quad y + 4z = 11 \quad \text{(Equation 5)}$$

• Step 3: Solve the new system of two equations:

  $$\begin{cases} -3y + z = 2 \\ y + 4z = 11 \end{cases}$$

  • Multiply the second equation by 3 to eliminate $y$:

    $$3y + 12z = 33$$

  • Add the two equations:

    $$(-3y + z) + (3y + 12z) = 2 + 33$$

    Simplify:

    $$13z = 35 \quad \Rightarrow \quad z = \frac{35}{13}$$

• Step 4: Substitute $z = \frac{35}{13}$ into one of the original equations (e.g., $y + 4z = 11$):

  $$y + 4\left(\frac{35}{13}\right) = 11 \quad \Rightarrow \quad y + \frac{140}{13} = 11 \quad \Rightarrow \quad y = 11 - \frac{140}{13} = \frac{143}{13} - \frac{140}{13} = \frac{3}{13}$$

• Step 5: Finally, substitute $y = \frac{3}{13}$ and $z = \frac{35}{13}$ into the equation $x = 6 - y - z$:

  $$x = 6 - \frac{3}{13} - \frac{35}{13} = 6 - \frac{38}{13} = \frac{78}{13} - \frac{38}{13} = \frac{40}{13}$$

Thus, the solution is:

2. Elimination Method

The elimination method involves adding or subtracting equations in such a way that one of the variables is eliminated. This method reduces the system to a simpler system, often starting with two equations in two unknowns.

Steps:

1. Multiply the equations (if necessary) so that the coefficients of one of the variables are opposites (i.e., one is the negative of the other).
2. Add or subtract the equations to eliminate one variable.
3. Solve the resulting system of two equations in two unknowns.
4. Once one variable is found, substitute it back into one of the original equations to solve for the other variables.

Example:

Solve the system:

• Step 1: Eliminate $z$ by adding the first and second equations:

  $$(x + y + z) + (2x - y + 3z) = 6 + 14 \quad \Rightarrow \quad 3x + 4z = 20 \quad \text{(Equation 4)}$$

• Step 2: Eliminate $z$ by adding the first and third equations:

  $$(x + y + z) + (3x + 2y - z) = 6 + 7 \quad \Rightarrow \quad 4x + 3y = 13 \quad \text{(Equation 5)}$$

• Step 3: Now solve the system of two equations:

  $$\begin{cases} 3x + 4z = 20 \\ 4x + 3y = 13 \end{cases}$$

  Solve this system using methods for two equations (substitution or elimination), and then substitute back to find the values of $x$, $y$, and $z$.

3. Matrix Method (Gaussian Elimination)

The matrix method, particularly Gaussian elimination, is an efficient algorithm for solving systems of linear equations. It involves transforming the system of equations into an augmented matrix and performing row operations to reduce the matrix to row echelon form or reduced row echelon form.

Steps:

1. Write the system of equations as an augmented matrix.
2. Use row operations (swap rows, multiply a row by a constant, add/subtract rows) to reduce the matrix to row echelon form.
3. Back-substitute to solve for the unknowns.

Types of Solutions

Like systems of two equations, a system of three equations can have:

• One solution (Unique Solution): The system is consistent and independent, and the three equations intersect at a single point.
• No solution: The system is inconsistent (the equations represent parallel planes that never meet).
• Infinite solutions: The system is dependent (the three equations represent the same plane or coincide in some way, giving infinitely many solutions).

Conclusion

Solving systems of three equations with three unknowns can be done using various methods, such as substitution, elimination, or matrix methods. While the substitution and elimination methods are often straightforward and intuitive, the matrix method (Gaussian elimination) provides a more systematic and efficient approach, especially for larger systems. Understanding the different methods and when to use each will help you handle complex systems of equations in algebra and other areas of mathematics.