📈College Algebra Unit 11 – Systems of Equations and Inequalities

Key Concepts

• Systems of equations involve multiple equations with shared variables that are solved simultaneously
• Linear systems consist of linear equations, meaning the variables are not raised to any power other than 1
• Solving systems of equations means finding the values of the variables that satisfy all equations in the system
• Graphing systems of equations can help visualize the solutions as the points of intersection between the lines or curves
• Substitution and elimination are two common algebraic methods for solving systems of equations
  • Substitution involves solving one equation for a variable and substituting that expression into the other equation
  • Elimination involves adding or subtracting equations to eliminate one of the variables
• Systems of equations can have one solution (consistent), no solution (inconsistent), or infinitely many solutions (dependent)

Types of Systems

• Two-variable systems involve two equations with two unknown variables, typically x and y
  • Example: $x + y = 5$ and $2x - y = 3$
• Three-variable systems involve three equations with three unknown variables, typically x, y, and z
  • Example: $x + y + z = 6$, $2x - y + z = 5$, and $x + 2y - z = 1$
• Homogeneous systems have all constant terms equal to zero, while non-homogeneous systems have at least one non-zero constant term
• Consistent systems have at least one solution, while inconsistent systems have no solutions
• Independent systems have a unique solution, while dependent systems have infinitely many solutions
• Over-determined systems have more equations than variables, while under-determined systems have fewer equations than variables

Solving Linear Systems

• Substitution method involves solving one equation for a variable and substituting the resulting expression into the other equation
  1. Solve one of the equations for one of the variables
  2. Substitute the expression from step 1 into the other equation
  3. Solve the resulting equation for the remaining variable
  4. Substitute the value from step 3 into the expression from step 1 to find the value of the other variable
• Elimination method involves multiplying equations by constants to eliminate one variable when the equations are added or subtracted
  1. Multiply one or both equations by constants to make the coefficients of one variable equal in magnitude but opposite in sign
  2. Add or subtract the equations to eliminate one variable
  3. Solve the resulting equation for the remaining variable
  4. Substitute the value from step 3 into one of the original equations to find the value of the other variable
• Gaussian elimination is a systematic approach to solving systems of equations using row operations to convert the system into row echelon form
• Cramer's rule uses determinants to solve systems of equations, but it is less efficient than substitution or elimination for larger systems

Graphing Systems

• Graphing systems of equations can help visualize the solutions as the points of intersection between the lines or curves
• For two-variable systems, the solutions are the coordinates of the points where the graphs of the equations intersect
  • One solution: the graphs intersect at a single point
  • No solution: the graphs are parallel and do not intersect
  • Infinitely many solutions: the graphs are coincident (overlapping)
• Graphing can be done by hand or using technology such as graphing calculators or online graphing tools
• Graphing is particularly useful for estimating solutions or checking the reasonableness of solutions obtained through other methods
• Graphing systems of inequalities involves shading the regions that satisfy all the inequalities simultaneously
  • The solution set is the intersection of the shaded regions for each inequality

Applications and Word Problems

• Systems of equations can model real-world situations involving multiple related quantities
• Word problems often require translating verbal descriptions into mathematical equations
  • Identify the unknown quantities and assign variables to them
  • Use the given information to set up equations expressing the relationships between the variables
• Common application areas include mixture problems, cost analysis, and geometric relationships
  • Mixture problems involve combining ingredients with different properties (e.g., price, concentration) to create a mixture with desired properties
  • Cost analysis problems involve determining prices, quantities, or costs based on given relationships (e.g., revenue, profit)
  • Geometric problems involve finding dimensions or angles based on given relationships (e.g., perimeter, area)
• Solving the system of equations provides the solution to the original word problem

Non-Linear Systems

• Non-linear systems involve equations with variables raised to powers other than 1 or containing functions such as exponentials, logarithms, or trigonometric functions
  • Example: $y = x^2$ and $y = 2x + 1$
• Solving non-linear systems often requires a combination of algebraic manipulation and graphical analysis
• Substitution and elimination methods can be used to simplify non-linear systems, but the resulting equations may still be non-linear
• Graphing non-linear systems can help identify the number and approximate location of solutions
• Some non-linear systems may have multiple solutions or no solutions, depending on the nature of the equations involved
• Numerical methods, such as Newton's method or the secant method, can be used to approximate solutions to non-linear systems

Inequalities and Their Systems

• Inequalities are mathematical statements that compare two expressions using symbols such as $<$, $>$, $\leq$, or $\geq$
• Solving inequalities involves finding the range of values that satisfy the inequality
  • Isolate the variable on one side of the inequality
  • Reverse the inequality sign if multiplying or dividing by a negative number
  • Express the solution using interval notation or graphing on a number line
• Systems of inequalities involve multiple inequalities that must be satisfied simultaneously
  • The solution set is the intersection of the solutions to each individual inequality
• Graphing systems of inequalities in two variables involves shading the regions that satisfy all the inequalities
  • Test points to determine which side of the boundary line to shade for each inequality
  • The solution set is the region where all the shaded areas overlap
• Linear programming is an application of systems of inequalities used to optimize a linear objective function subject to linear constraints

Advanced Techniques and Special Cases

• Matrix methods can be used to solve systems of equations by representing the system as a matrix equation and using row operations to solve
  • Augmented matrix notation: $[A|b]$, where $A$ is the coefficient matrix and $b$ is the constant vector
  • Row echelon form: a matrix where all entries below the main diagonal are zero and the leading entry in each row is 1
• Partial fraction decomposition can be used to solve systems of equations involving rational functions by expressing the solution as a sum of simpler fractions
• Eigenvalues and eigenvectors can be used to analyze the behavior of systems of linear differential equations
• Singular systems have a coefficient matrix with a determinant equal to zero, indicating either no solution or infinitely many solutions
• Ill-conditioned systems are sensitive to small changes in the coefficients or constant terms, leading to significant changes in the solution
• Iterative methods, such as Jacobi iteration or Gauss-Seidel iteration, can be used to approximate solutions to large systems of equations