Honors Pre-Calculus Unit 9 Review: Systems of Equations and Inequalities

Key Concepts

• Systems of equations involve multiple equations with multiple variables that are solved simultaneously
• Inequalities can also be combined into systems, where the solution set satisfies all inequalities
• The number of equations in a system should match the number of variables to find a unique solution
• Systems can have one solution (consistent), no solution (inconsistent), or infinitely many solutions (dependent)
• Equivalent systems have the same solution set but may have different equations
• Substitution and elimination are common algebraic methods for solving systems
• Graphing is a visual method for solving systems, where the point of intersection is the solution

Types of Systems

• Linear systems consist of first-degree equations and produce straight lines when graphed
  • Example: $2x + 3y = 7$ and $x - y = 1$
• Nonlinear systems involve equations with higher degrees (quadratic, cubic, etc.) and produce curves
  • Example: $y = x^2$ and $y = -x + 5$
• Homogeneous systems have all constant terms equal to zero, while non-homogeneous systems have non-zero constants
• Consistent systems have at least one solution, while inconsistent systems have no solution
• Dependent systems have infinitely many solutions, as the equations are multiples of each other
• Systems of inequalities involve inequalities ($<, >, \leq, \geq$) instead of equations ($=$)
  • Example: $y < 2x + 1$ and $y \geq -x + 3$

Solving Methods

• Substitution involves solving one equation for a variable and substituting the result into the other equation
  • Solve for the simplest variable and substitute to eliminate a variable
• Elimination (addition) method involves adding or subtracting equations to eliminate a variable
  • Multiply equations by constants to make coefficients of one variable opposites, then add to eliminate
• Graphing method involves plotting the equations and finding the point(s) of intersection
  • The coordinates of the intersection point(s) are the solution(s) to the system
• Matrix methods (Cramer's Rule, inverse matrices) can be used for larger systems
• For systems of inequalities, graph each inequality and find the region that satisfies all inequalities
  • Shade the overlapping region that satisfies all inequalities

Graphical Representations

• Each equation in a system represents a line (linear) or curve (nonlinear) when graphed
• The point(s) where the lines or curves intersect represent the solution(s) to the system
  • One intersection point indicates a unique solution
  • No intersection points indicate an inconsistent system with no solution
  • Overlapping lines indicate a dependent system with infinitely many solutions
• For systems of inequalities, each inequality divides the plane into two regions (satisfied and unsatisfied)
  • The solution set is the region that satisfies all inequalities simultaneously
  • Shade the overlapping region that satisfies all inequalities
• Graphing can provide a visual understanding of the system and its solution(s)

Applications and Word Problems

• Systems of equations can model real-world situations involving multiple variables
  • Example: Pricing plans for a cell phone company based on minutes and data usage
• Identify the variables in the problem and assign them symbols (e.g., $x$, $y$)
• Write equations or inequalities based on the given information and relationships
  • Example: If one pound of apples costs $2 and one pound of bananas costs $1.50, and you spend $10 on 6 pounds of fruit, the system would be:
    • $2a + 1.5b = 10$
    • $a + b = 6$
• Solve the system using an appropriate method (substitution, elimination, graphing)
• Interpret the solution in the context of the original problem
  • Example: The solution $a = 2, b = 4$ means 2 pounds of apples and 4 pounds of bananas were purchased

Common Pitfalls and Tips

• Ensure that the number of equations matches the number of variables for a unique solution
• Check that equations are properly aligned when using the elimination method
• Be careful with signs when adding or subtracting equations in the elimination method
• When substituting, make sure to substitute the entire expression, not just the variable
• Check the solution by substituting it back into the original equations
• For systems of inequalities, pay attention to the direction of the inequality signs
• Remember that the solution to a system of inequalities is a region, not a single point
• Double-check the feasibility of the solution in the context of the word problem
  • Example: Negative solutions may not make sense for quantities like prices or distances

Advanced Techniques

• Gaussian elimination is a systematic method for solving larger systems of equations
  • Involves converting the system into row echelon form through row operations
• Cramer's Rule uses determinants to solve systems with unique solutions
  • Useful when the system has the same number of equations as variables
• Inverse matrix method involves expressing the system in matrix form $AX = B$ and solving using $X = A^{-1}B$
  • Requires the coefficient matrix $A$ to be square and invertible
• Partial fractions decomposition can be used to solve systems involving rational functions
  • Decompose rational expressions into simpler fractions and solve the resulting system
• Substitution can be used to solve systems of nonlinear equations by solving for one variable in terms of the other
  • Example: For $y = x^2$ and $y = -x + 5$, solve the second equation for $x$ and substitute into the first

Practice Problems

1. Solve the system: $3x - 2y = 8$ and $2x + y = 7$
2. Solve the system: $y = x^2 - 4$ and $y = 2x + 1$
3. Solve the system of inequalities: $y \leq 3x + 2$ and $y > -x + 1$
4. A store sells two types of candy. Candy A costs $1.50 per pound, and Candy B costs $2 per pound. If the store sold 10 pounds of candy for $18, how many pounds of each type were sold?
5. Solve the system using Cramer's Rule: $2x + 3y = 7$ and $x - y = 1$
6. Solve the system using the inverse matrix method: $3x - 2y = 5$ and $2x + y = 4$
7. Solve the system of nonlinear equations: $y = x^3 - 1$ and $y = -x + 2$
8. Solve the system using substitution: $4x - 3y = 11$ and $2x + y = 5$