Systems of Equations - Formulas & Methods
IB Mathematics Analysis & Approaches (SL & HL)
📐 General Forms
2×2 System (Two equations, two unknowns):
\[\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\]
Geometrically: Represents two lines in 2D space
3×3 System (Three equations, three unknowns):
\[\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}\]
Geometrically: Represents three planes in 3D space
🔢 Matrix Representation
Matrix Form:
A system of linear equations can be written as a matrix equation:
\[AX = B\]
where \(A\) is the coefficient matrix, \(X\) is the variable column matrix, \(B\) is the constant column matrix
For a 2×2 System:
\[\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\]
For a 3×3 System:
\[\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}\]
🔄 Inverse Matrix Method
Solution Formula:
If the coefficient matrix \(A\) is invertible (det\(A \neq 0\)), the solution is:
\[X = A^{-1}B\]
2×2 Matrix Inverse:
For matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\):
\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]
where \(ad - bc\) is the determinant of \(A\)
Steps to Solve:
1. Write the system in matrix form \(AX = B\)
2. Find the inverse matrix \(A^{-1}\) (if it exists)
3. Multiply both sides: \(X = A^{-1}B\)
4. Calculate the result to find values of variables
📊 Cramer's Rule
When to Use:
Cramer's Rule is used when the system has a unique solution (det\(A \neq 0\)). It expresses solutions in terms of determinants.
For a 2×2 System:
\[x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}\]
where:
\(\det(A) = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\) (original determinant)
\(\det(A_x) = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}\) (replace x-column with constants)
\(\det(A_y) = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}\) (replace y-column with constants)
For a 3×3 System:
\[x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)}\]
Replace each variable's column with the constants column to get the respective numerator determinant
🔀 Gaussian Elimination (Row Reduction)
Augmented Matrix:
Combine the coefficient matrix and constant matrix into an augmented matrix:
\[\left[\begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{array}\right]\]
Row Operations:
1. Row Switching: \(R_i \leftrightarrow R_j\) (swap two rows)
2. Row Multiplication: \(kR_i \to R_i\) (multiply row by non-zero constant)
3. Row Addition: \(R_i + kR_j \to R_i\) (add multiple of one row to another)
Row Echelon Form (REF):
Transform to upper triangular form with 1's on the diagonal:
\[\left[\begin{array}{ccc|c} 1 & * & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \end{array}\right]\]
Then use back-substitution to find the solution
🔄 Substitution Method
Steps:
1. Solve one equation for one variable in terms of the other(s)
2. Substitute this expression into the other equation(s)
3. Solve the resulting equation(s)
4. Back-substitute to find all variable values
Best Used When:
• One variable has a coefficient of 1 or -1
• One equation is already solved for a variable
• Working with simple 2×2 systems
➖ Elimination Method
Steps:
1. Multiply equations by constants to make coefficients of one variable equal
2. Add or subtract equations to eliminate that variable
3. Solve the resulting equation for the remaining variable
4. Substitute back to find the other variable(s)
Best Used When:
• Coefficients are easily made equal
• Working with 2×2 systems
• Variables can be quickly eliminated
🔍 Types of Solutions
1. Unique Solution (One Solution):
Condition: \(\det(A) \neq 0\) (matrix is invertible)
Geometric interpretation: Lines/planes intersect at exactly one point
Result: One set of values for all variables
2. Infinitely Many Solutions:
Condition: \(\det(A) = 0\) and equations are dependent (consistent)
Geometric interpretation: Lines/planes coincide or overlap
Result: Solutions expressed in terms of a parameter
3. No Solution:
Condition: \(\det(A) = 0\) and equations are inconsistent
Geometric interpretation: Lines/planes are parallel (never intersect)
Result: System is inconsistent, leads to contradiction (e.g., 0 = 5)
📐 Determinants
2×2 Determinant:
\[\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\]
3×3 Determinant (Expansion along first row):
\[\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix}\]
Simplified: \(a(ei - fh) - b(di - fg) + c(dh - eg)\)
⚠️ Special Cases
Homogeneous System:
\[AX = 0\]
Always has at least the trivial solution \(X = 0\). Has non-trivial solutions if \(\det(A) = 0\)
Underdetermined System:
Fewer equations than unknowns → Infinitely many solutions or no solution
🧮 Calculator & Technology Tips
📱 Using GDC/Calculator:
• Most calculators have a built-in system solver in the equation or algebra menu
• Can calculate matrix inverse directly
• Can compute determinants automatically
• Can perform row operations on augmented matrices
• TI Calculators: APPS → PolySmtl or use matrix operations
• Casio Calculators: MENU → Equation/System
• Always write your method algebraically if required by the question
💡 Strategy Tip: For 2×2 systems, use substitution or elimination. For 3×3 systems, use Gaussian elimination, matrix inverse method, or your GDC. Always check if \(\det(A) = 0\) to determine the type of solution before solving.