Systems with Three Variables - Algebra-2

1. Fundamental Concepts

Definition: A system with three variables consists of three equations involving three unknowns, typically represented as $x$, $y$, and $z$.
Solution Set: The solution to a system of three equations is the set of values $(x, y, z)$ that satisfy all three equations simultaneously.
Consistency: A system is consistent if it has at least one solution; otherwise, it is inconsistent.

2. Key Concepts

Elimination Method: $$a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3$$ - Use addition or subtraction to eliminate one variable.

Substitution Method: Solve one equation for one variable and substitute into the other equations.

Matrix Method (Gaussian Elimination): Use row operations to transform the augmented matrix into row-echelon form.

3. Examples

Example 1 (Basic)

Problem: Solve the system: $$ \begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ 3x + y - z = 4 \end{cases} $$

Step-by-Step Solution:

Add the first and second equations to eliminate $y$: $$ (x + y + z) + (2x - y + z) = 6 + 3 \\ 3x + 2z = 9 $$
Add the first and third equations to eliminate $y$: $$ (x + y + z) + (3x + y - z) = 6 + 4 \\ 4x = 10 \\ x = \frac{10}{4} = 2.5 $$
Substitute $x = 2.5$ into $3x + 2z = 9$: $$ 3(2.5) + 2z = 9 \\ 7.5 + 2z = 9 \\ 2z = 1.5 \\ z = 0.75 $$
Substitute $x = 2.5$ and $z = 0.75$ into $x + y + z = 6$: $$ 2.5 + y + 0.75 = 6 \\ y = 6 - 3.25 \\ y = 2.75 $$

Validation: Substitute $x = 2.5$, $y = 2.75$, $z = 0.75$ into the original equations: $$ 2.5 + 2.75 + 0.75 = 6 \\ 2(2.5) - 2.75 + 0.75 = 3 \\ 3(2.5) + 2.75 - 0.75 = 4 $$ All equations are satisfied.

Example 2 (Intermediate)

Problem: Solve the system: $$ \begin{cases} x + 2y - z = 5 \\ 2x - y + 3z = 1 \\ 3x + y - 2z = 4 \end{cases} $$

Step-by-Step Solution:

Multiply the first equation by 2 and subtract from the second equation to eliminate $x$: $$ 2(x + 2y - z) = 2(5) \\ 2x + 4y - 2z = 10 \\ (2x - y + 3z) - (2x + 4y - 2z) = 1 - 10 \\ -5y + 5z = -9 \\ y - z = \frac{9}{5} $$
Multiply the first equation by 3 and subtract from the third equation to eliminate $x$: $$ 3(x + 2y - z) = 3(5) \\ 3x + 6y - 3z = 15 \\ (3x + y - 2z) - (3x + 6y - 3z) = 4 - 15 \\ -5y + z = -11 \\ y - z = \frac{11}{5} $$
Solve the system of two equations in $y$ and $z$: $$ \begin{cases} y - z = \frac{9}{5} \\ y - z = \frac{11}{5} \end{cases} $$ This system is inconsistent, indicating no solution.

Validation: The system is inconsistent, so there is no solution.

4. Problem-Solving Techniques

Elimination Strategy: Use addition or subtraction to eliminate one variable at a time.
Substitution Strategy: Solve one equation for one variable and substitute into the other equations.
Matrix Strategy (Gaussian Elimination): Use row operations to transform the augmented matrix into row-echelon form.
Error-Proofing: Always check the solution by substituting back into the original equations.