Solving Simultaneous Equations

• Simultaneous equations involve two or more equations with the same variables
• Solutions to simultaneous equations are the variable values satisfying all equations
• Solving simultaneous equations determines the common solution set for all equations

Linear Simultaneous Equations

• Linear simultaneous equations have a variable with the highest power of 1
• Linear simultaneous equations represent straight lines when graphed
• The solution to a system of two linear equations is the intersection point of the two lines
• No solution exists if the lines are parallel
• Infinitely many solutions exist if the lines are the same

Methods to Solve Linear Simultaneous Equations

• Graphical Method: graph each equation to find the intersection point
• This method is useful for visualizing the solution however can be less accurate
• Substitution Method: solve one equation to express one variable in terms of the other, then substitute
• This method reduces the system to a single equation in one variable
• Solve for the remaining variable and substitute back to find the value of the other variable
• Elimination Method: multiply equations by constants so one variable's coefficients are opposites
• Add the equations to eliminate that variable
• Solve for the remaining variable and substitute back to find the eliminated variable's value

Example: Solving by Substitution

• System: x + y = 5 and 2x - y = 1
• First equation solved for y: y = 5 - x
• Substitute into the second equation: 2x - (5 - x) = 1
• Simplify and solve for x: 2x - 5 + x = 1 => 3x = 6 => x = 2
• Substitute x = 2 back into y = 5 - x: y = 5 - 2 = 3
• The solution is x = 2, y = 3

Example: Solving by Elimination

• System: 3x + 2y = 7 and 4x - 2y = 0
• The coefficients of y are already opposites
• Add the equations: (3x + 2y) + (4x - 2y) = 7 + 0 => 7x = 7 => x = 1
• Substitute x = 1 into the first equation: 3(1) + 2y = 7 => 2y = 4 => y = 2
• The solution is x = 1, y = 2

Non-Linear Simultaneous Equations

• Non-linear simultaneous equations involve variables with powers greater than 1, and other non-linear functions
• Non-linear simultaneous equations can represent curves like circles, parabolas, hyperbolas, etc
• Solving non-linear systems is more complex

Methods to Solve Non-Linear Simultaneous Equations

• Substitution Method: solve one equation for one variable in terms of the other, then substitute
• This method often produces a more complex equation to solve
• Elimination Method: manipulation can eliminate terms, however this method is less common
• Graphical Method: graphing helps visualize solutions, but may not provide exact answers

Example: Solving a Non-Linear System

• System: x^2 + y^2 = 25 and y = x + 1
• Substitute y = x + 1 into the first equation: x^2 + (x + 1)^2 = 25
• Expand and simplify: x^2 + x^2 + 2x + 1 = 25 => 2x^2 + 2x - 24 = 0
• Divide by 2: x^2 + x - 12 = 0
• Factor: (x + 4)(x - 3) = 0
• Therefore x = -4 or x = 3
• If x = -4, y = -4 + 1 = -3
• If x = 3, y = 3 + 1 = 4
• The solutions are (-4, -3) and (3, 4)

Applications of Simultaneous Equations

• Solving word problems involving multiple unknowns
• Modeling real-world scenarios in science, engineering, and economics
• Finding equilibrium points in supply and demand curves
• Circuit analysis in electrical engineering

Solving Systems with Three or More Variables

• Use substitution or elimination to reduce the system to fewer variables
• Matrix methods like Gaussian elimination and row reduction are used for linear systems
• Software and calculators are helpful for solving larger systems

Matrix Representation of Linear Systems

• A system of linear equations can be in matrix form as AX = B
• A is the coefficient matrix
• X is the variable matrix (column vector)
• B is the constant matrix (column vector)
• Matrix methods like Gaussian elimination or the inverse of A (if it exists) can be used to solve for X

Determinants and Cramer's Rule

• Cramer's Rule: A method for solving linear systems using determinants
• This method can be computationally intensive for large systems
• Determinants determine if a system has a unique solution
• A non-zero determinant of the coefficient matrix produces a unique solution
• A zero determinant produces either no solution or infinitely many solutions

Homogeneous Systems

• A homogeneous system is one where all constant terms are zero (AX = 0)
• Homogeneous systems always have the trivial solution (X = 0)
• Non-trivial solutions exist if the determinant of A is zero

Consistency and Independence

• Consistent System: A system that has at least one solution
• Inconsistent System: A system that has no solution
• Independent Equations: Equations that give unique information; no equation can be derived from the others
• Dependent Equations: Equations where one can be derived from another; they provide redundant information

Number of Solutions

• A system of linear equations can have:
• A unique solution (lines intersect at one point)
• No solution (lines are parallel)
• Infinitely many solutions (lines overlap)
• The number of solutions is determined by examining the determinant of the coefficient matrix and the system's consistency