Simultaneous Linear Equations (3 Variables)

Simultaneous Linear Equations (3 Variables)#

Simultaneous Linear Equations (2 Variables) Revision#

Theory#

Before tackling three-variable systems, let’s review two-variable systems. A system of two linear equations in two variables:

\[\begin{split}\begin{align} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{align}\end{split}\]

has three possible outcomes:

  • Unique solution: Lines intersect at one point

  • No solution: Lines are parallel

  • Infinitely many solutions: Lines are identical

The main solution methods are substitution and elimination.

Application#

Examples#

Example 1#

Solve: \(\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}\)

Method 1: Elimination

\(\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases} \quad \text{(Add equations to eliminate y)}\)

\(3x = 9 \quad \text{(Solve for x)}\)

\(x = 3 \quad \text{(Substitute back)}\)

\(3 - y = 2\), so \(y = 1 \quad \text{(Solution: (3, 1))}\)

Interactive Visualization: Two Variable System Review#

Multiple Choice Questions#

Simultaneous Linear Equations (3 Variables)#

Theory#

A system of three linear equations in three variables takes the form:

\[\begin{split}\begin{align} a_1x + b_1y + c_1z &= d_1 \\ a_2x + b_2y + c_2z &= d_2 \\ a_3x + b_3y + c_3z &= d_3 \end{align}\end{split}\]

Geometrically, each equation represents a plane in 3D space. The solution is the point where all three planes intersect.

Possible outcomes:

  1. Unique solution: Three planes intersect at one point

  2. No solution: Planes don’t have a common intersection point

  3. Infinitely many solutions: Planes intersect along a line or are identical

Solution methods:

  1. Elimination Method: Systematically eliminate variables

  2. Substitution Method: Express variables in terms of others

  3. Matrix Methods: Gaussian elimination, Cramer’s rule

Tip

When solving 3-variable systems, organize your work systematically. Label your equations and keep track of which variables you’re eliminating at each step.

Interactive Visualization: Three Variable System Concept#

Application#

Examples#

Example 1#

Solve by elimination: \(\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 1 \end{cases}\)

Method 1: Systematic Elimination

\(\begin{cases} x + y + z = 6 \quad (1) \\ 2x - y + z = 3 \quad (2) \\ x + 2y - z = 1 \quad (3) \end{cases} \quad \text{(Label equations)}\)

\((2) - (1): x - 2y = -3 \quad (4) \quad \text{(Eliminate z from equations 1 and 2)}\)

\((1) + (3): 2x + 3y = 7 \quad (5) \quad \text{(Eliminate z from equations 1 and 3)}\)

From (4): \(x = 2y - 3 \quad \text{(Substitute into equation 5)}\)

\(2(2y - 3) + 3y = 7 \quad \text{(Solve for y)}\)

\(4y - 6 + 3y = 7\), so \(7y = 13\), thus \(y = \frac{13}{7} \quad \text{(Find x and z)}\)

\(x = 2(\frac{13}{7}) - 3 = \frac{5}{7}\), and \(z = 6 - \frac{5}{7} - \frac{13}{7} = \frac{24}{7} \quad \text{(Solution: \)(\frac{5}{7}, \frac{13}{7}, \frac{24}{7})\()}\)

Example 2#

Solve: \(\begin{cases} 2x + y - z = 4 \\ x - y + 2z = 1 \\ 3x + 2y + z = 7 \end{cases}\)

Method 1: Elimination Strategy

\(\begin{cases} 2x + y - z = 4 \quad (1) \\ x - y + 2z = 1 \quad (2) \\ 3x + 2y + z = 7 \quad (3) \end{cases} \quad \text{(Original system)}\)

\((1) + (2): 3x + z = 5 \quad (4) \quad \text{(Eliminate y)}\)

\(2 \cdot (2) + (1): 4x + 3z = 6 \quad (5) \quad \text{(Another equation without y)}\)

From (4): \(z = 5 - 3x \quad \text{(Substitute into equation 5)}\)

\(4x + 3(5 - 3x) = 6 \quad \text{(Solve for x)}\)

\(4x + 15 - 9x = 6\), so \(-5x = -9\), thus \(x = \frac{9}{5} \quad \text{(Find remaining variables)}\)

\(z = 5 - 3(\frac{9}{5}) = \frac{-2}{5}\), and \(y = 4 - 2(\frac{9}{5}) - (-\frac{2}{5}) = 2 \quad \text{(Solution: \)(\frac{9}{5}, 2, -\frac{2}{5})\()}\)

Example 3#

Identify system type: \(\begin{cases} x + y + z = 3 \\ 2x + 2y + 2z = 6 \\ x - y + z = 1 \end{cases}\)

Method 1: Analysis

\(\begin{cases} x + y + z = 3 \quad (1) \\ 2x + 2y + 2z = 6 \quad (2) \\ x - y + z = 1 \quad (3) \end{cases} \quad \text{(Notice equation 2 is 2 × equation 1)}\)

Equation (2) is redundant, so we have only two independent equations for three variables.

\((1) - (3): 2y = 2\), so \(y = 1 \quad \text{(From equation 1: \)x + z = 2\()}\)

This gives infinitely many solutions of the form \((t, 1, 2-t)\) where \(t\) is any real number.

Multiple Choice Questions#

Sector Specific Questions: Three Variable Systems Applications#

Key Takeaways#

Important

  1. A system of 3 linear equations in 3 variables generally requires 3 independent equations for a unique solution

  2. Geometrically, each equation represents a plane in 3D space

  3. Solution types: unique point, no solution, or infinitely many solutions (line or plane)

  4. Systematic elimination is the most reliable solution method

  5. Always check that equations are independent (not multiples of each other)

  6. Label equations clearly and organize your elimination steps

  7. Verify solutions by substituting back into all original equations

  8. Real-world applications often involve resource allocation, network analysis, and mixture problems

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