📐Algebra and Trigonometry Unit 11 – Systems of Equations & Inequalities

Key Concepts

• Systems of equations involve multiple equations with shared variables that must be solved simultaneously
• Inequalities express a range of values rather than a specific solution and can also be combined into systems
• Linear systems consist of first-degree equations while nonlinear systems include higher-degree terms
• Substitution and elimination are two primary algebraic methods for solving systems of equations
  • Substitution involves isolating a variable in one equation and plugging it into the other
  • Elimination adds or subtracts equations to cancel out a variable
• Graphing systems of equations reveals the point(s) of intersection which represent the solution(s)
• Matrices can be used to solve systems of equations through row reduction and Gaussian elimination
• Systems of inequalities are solved graphically, with overlapping shaded regions indicating the solution set
• Applications of systems range from simple cost problems to complex modeling in physics and engineering

Types of Systems

• Two-variable systems are the most common, consisting of two equations with variables $x$ and $y$
  • Example: $x + y = 10$ and $2x - y = 4$
• Three-variable systems introduce a third variable $z$ and require three equations to solve
  • Example: $x + y + z = 6$, $2x - y + z = 5$, and $x - 2y - z = 1$
• Nonlinear systems include equations with higher powers, radicals, or trigonometric functions
  • Example: $x^2 + y^2 = 25$ and $y = \sin(x)$
• Homogeneous systems have a constant term of zero in every equation, always including the trivial solution $(0, 0)$
• Inconsistent systems have no solution, represented graphically by parallel lines that never intersect
• Dependent systems have infinitely many solutions, represented by overlapping lines or planes

Solving Linear Systems

• Substitution method isolates one variable to plug into the other equation(s)
  1. Solve one equation for a variable (e.g., $y = 3x + 2$)
  2. Substitute the expression into the other equation and solve for the remaining variable
  3. Back-substitute the value to find the other variable(s)
• Elimination method adds or subtracts equations to cancel out a variable
  1. Multiply equations by constants to make coefficients of one variable opposites
  2. Add the equations to eliminate the variable
  3. Solve the resulting equation and back-substitute to find the other variable(s)
• Matrices can solve systems using augmented matrices and row reduction
  • Augmented matrix combines the coefficient matrix and constant vector
  • Row reduction uses elementary row operations to convert the matrix to row echelon form
  • Back-substitution reveals the solution from the row-reduced matrix

Graphing Systems

• Graphing linear systems plots each equation as a line, with the point(s) of intersection representing the solution(s)
  • Consistent systems have one solution (intersecting lines) or infinitely many (overlapping lines)
  • Inconsistent systems have no solution, shown by parallel lines that never intersect
• Graphing nonlinear systems can result in more complex curves and multiple points of intersection
  • Example: Graphing a circle ($x^2 + y^2 = 25$) and sine wave ($y = \sin(x)$) yields two solutions
• Graphing inequalities shades the region satisfying the condition, with solid lines for $\leq$ or $\geq$ and dashed for $<$ or $>$
• Systems of inequalities are solved by graphing each inequality and finding the overlapping shaded region(s)
  • The solution set includes all points within the overlapping region(s)

Applications in Real Life

• Cost and revenue problems often use systems to find break-even points or optimize profit
  • Example: A company sells shirts and hats. Shirts cost $10 to make and sell for$25, while hats cost $5 and sell for$15. If they have a budget of $1000 and want to make at least$2000 in revenue, a system can determine how many of each to produce.
• Physics problems use systems to model motion, forces, and energy
  • Example: A boat crosses a river with a current. The boat's velocity and the current's velocity can be represented as vectors, forming a system to determine the boat's resultant path.
• Chemistry problems involve balancing reactions and solving concentration mixtures
  • Example: A chemist has a 30% acid solution and a 50% solution. A system can determine how much of each to mix to obtain a desired volume and concentration.
• Geometry problems apply systems to find measurements of shapes and angles
  • Example: Finding the dimensions of a rectangle given its perimeter and area forms a system of equations.

Inequalities and Their Systems

• Inequalities use $<$, $>$, $\leq$, or $\geq$ to express a range of values rather than a specific solution
  • Example: $2x - 3 < 7$ represents all values of $x$ less than 5
• Graphing inequalities shades the region satisfying the condition, with the line type (solid or dashed) determined by the inequality symbol
• Systems of inequalities combine multiple inequalities to find a common solution set
  • The solution set is represented graphically by the overlapping shaded regions
  • Example: $y > x + 1$ and $y < 2x - 3$ form a system with a solution set bounded by the two lines
• Linear programming optimizes an objective function subject to a system of constraints (inequalities)
  • The constraints form a feasible region, with the optimal solution occurring at a vertex or along an edge
  • Applies to problems in business, economics, and resource allocation

Advanced Techniques

• Cramer's Rule uses determinants to solve systems of equations
  • For a 2x2 system, $x = \frac{|A_x|}{|A|}$ and $y = \frac{|A_y|}{|A|}$, where $A$ is the coefficient matrix and $A_x$, $A_y$ replace a column of $A$ with the constant terms
  • Extends to larger systems using higher-order determinants
• Partial fractions decomposition breaks down rational expressions into simpler terms
  • Useful for solving systems involving fractions or integrating rational functions
  • Example: $\frac{2x+3}{(x-1)(x+2)}$ can be decomposed into $\frac{A}{x-1} + \frac{B}{x+2}$
• Nonlinear systems can be solved using substitution, graphing, or advanced methods like Newton's method or the Jacobian matrix
  • Newton's method iteratively approximates solutions using derivatives and linear approximations
  • The Jacobian matrix generalizes Newton's method to higher dimensions

Common Pitfalls and Tips

• Always check the solution by plugging it back into the original equations
• Be careful when multiplying or dividing inequalities by negative numbers, as it reverses the inequality sign
• When graphing systems, ensure the scales of the axes are consistent and appropriate for the problem
• Label graphs clearly, including the equations, solution points, and shaded regions for inequalities
• Double-check the algebra when solving, especially signs and distribution of terms
• Consider the context of the problem when interpreting solutions, discarding extraneous or nonsensical results
  • Example: A negative number of items produced or time elapsed may be mathematically valid but practically impossible
• Recognize special cases, such as inconsistent or dependent systems, to avoid wasted effort trying to solve them algebraically
• Practice various techniques to build fluency and identify the most efficient method for a given problem