AP Precalculus: Systems of Equations Formulas & Methods

1. System Classification

• Consistent & Independent: One unique solution (lines intersect at one point)
• Consistent & Dependent: Infinite solutions (same line)
• Inconsistent: No solution (parallel lines)

2. Solving by Graphing

• Graph both equations on same coordinate plane
• Solution = intersection point(s) \((x, y)\)
• For system: \(\begin{cases} y = m_1x + b_1 \\ y = m_2x + b_2 \end{cases}\)
• If \(m_1 = m_2\) and \(b_1 = b_2\): infinite solutions
• If \(m_1 = m_2\) and \(b_1 \neq b_2\): no solution

3. Substitution Method

• Solve one equation for one variable: \(y = f(x)\)
• Substitute into the other equation
• Solve for remaining variable, then back-substitute
• Example: \(\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}\)
  • From first: \(y = 5-x\)
  • Substitute: \(2x - (5-x) = 1 \Rightarrow x = 2, y = 3\)

4. Elimination Method

• Multiply equations to create opposite coefficients
• Add/subtract to eliminate one variable
• Solve for remaining variable, then substitute back
• Example: \(\begin{cases} 2x + 3y = 7 \\ 4x - 3y = 5 \end{cases}\)
  • Add equations: \(6x = 12 \Rightarrow x = 2\)
  • Back-substitute: \(y = 1\)

5. Augmented Matrices (Row Operations)

• Write system as augmented matrix: \(\left[\begin{array}{cc|c} a & b & e \\ c & d & f \end{array}\right]\)
• Row operations: Swap rows, multiply row by constant, add multiple of one row to another
• Goal: Reduced row echelon form (RREF)
• Example: \(\begin{cases} x + 2y = 3 \\ 2x + y = 4 \end{cases} \Rightarrow \left[\begin{array}{cc|c} 1 & 2 & 3 \\ 2 & 1 & 4 \end{array}\right]\)

6. Three Variables: Substitution

• System: \(\begin{cases} ax + by + cz = d \\ ex + fy + gz = h \\ ix + jy + kz = l \end{cases}\)
• Solve one equation for one variable, substitute into others
• Reduce to 2-variable system, then solve

7. Three Variables: Elimination

• Use pairs of equations to eliminate same variable
• Create two 2-variable equations, solve that system
• Back-substitute to find third variable

8. Number of Solutions (3 Variables)

• One solution: Three planes intersect at one point
• No solution: Planes are parallel or form no common intersection
• Infinite solutions: Planes intersect along a line or are identical