Systems of linear equations and inequalities are powerful tools for solving complex problems. They allow us to model relationships between variables and find solutions that satisfy multiple conditions simultaneously. This topic builds on our understanding of linear equations and inequalities, extending it to handle more intricate scenarios.

In this section, we'll learn different methods for solving systems, including graphical, substitution, and elimination approaches. We'll also explore how to graph systems of inequalities and apply these concepts to real-world situations, connecting abstract math to practical problem-solving.

Solving Systems of Equations

Graphical Method

• Involves graphing both equations on the same coordinate plane
• The solution is the point(s) where the lines intersect
• The x and y coordinates of the point(s) of intersection are the solution to the system

Substitution Method

• Solve one equation for one of the variables
• Substitute that expression into the other equation to solve for the remaining variable
• Substitute the values obtained back into either of the original equations to find the corresponding value of the other variable
• Example:
  • Given the system of equations:
    • $2x + y = 7$
    • $x - y = 1$
  • Solve the first equation for y: $y = 7 - 2x$
  • Substitute this expression into the second equation: $x - (7 - 2x) = 1$
  • Solve for x: $x = 4$
  • Substitute x = 4 back into the first equation to solve for y: $2(4) + y = 7$, so $y = -1$
  • The solution is (4, -1)

Elimination Method

• Also known as the addition method
• Multiply one or both equations by a constant to eliminate one of the variables when the equations are added together
• Results in an equation with only one variable, which can be solved
• Substitute the value obtained back into one of the original equations to solve for the other variable
• If the coefficients of one variable are opposites, add the equations together to eliminate that variable
• If the coefficients are not opposites, multiply one or both equations by a constant to make the coefficients opposites
• Example:
  • Given the system of equations:
    • $3x + 2y = 11$
    • $2x - y = 1$
  • Multiply the second equation by 2 to make the coefficients of y opposites: $4x - 2y = 2$
  • Add the equations together to eliminate y: $7x = 13$
  • Solve for x: $x = \frac{13}{7}$
  • Substitute x = $\frac{13}{7}$ back into the first equation to solve for y: $3(\frac{13}{7}) + 2y = 11$, so $y = \frac{4}{7}$
  • The solution is ($\frac{13}{7}$, $\frac{4}{7}$)

Number of Solutions for Systems

One Solution (Consistent and Independent)

• Lines intersect at a single point
• Slopes of the lines are different
• y-intercepts of the lines are different

No Solution (Inconsistent)

• Lines are parallel and have different y-intercepts
• Slopes of the lines are the same, but the y-intercepts are different
• When solving algebraically, the result is a false statement (e.g., 0 = 1)

Infinitely Many Solutions (Consistent and Dependent)

• Lines are coincident (overlapping)
• Slopes and y-intercepts of the lines are the same
• When solving algebraically, the result is a true statement (e.g., 0 = 0)
• Example:
  • Given the system of equations:
    • $2x + 3y = 6$
    • $4x + 6y = 12$
  • The second equation is a multiple of the first equation, so the lines are coincident
  • The system has infinitely many solutions, which are all the points on the line $2x + 3y = 6$

Graphing Systems of Inequalities

Graphing a Linear Inequality

• Graph the corresponding equation as a dashed line if the inequality is strict (< or >) or a solid line if the inequality is not strict (≤ or ≥)
• Shade the half-plane above the line if the y-value is greater than or equal to the equation, or shade the half-plane below the line if the y-value is less than or equal to the equation

Identifying the Solution Set

• Graph each inequality separately
• Identify the region where all the shaded half-planes overlap
• The overlapping region is the solution set of the system
• The solution set can be bounded (a finite region), unbounded (an infinite region), or empty (no solution)
• To determine the coordinates of the vertices of the solution set, solve the corresponding system of linear equations formed by the boundary lines of the inequalities
• Example:
  • Given the system of inequalities:
    • $y ≤ 2x + 1$
    • $y > -x + 3$
  • Graph the first inequality as a solid line and shade the half-plane below the line
  • Graph the second inequality as a dashed line and shade the half-plane above the line
  • The solution set is the region where the shaded half-planes overlap, which is an unbounded region

Real-World Applications of Systems

Modeling with Systems of Linear Equations

• Identify the variables and define what they represent
• Formulate the equations based on the given information and constraints
• Examples:
  • Finding the break-even point for a business selling multiple products
  • Determining the equilibrium price and quantity in a market with supply and demand equations
  • Calculating the concentrations of chemical solutions in a mixture

Modeling with Systems of Linear Inequalities

• Identify the variables and define what they represent
• Formulate the inequalities based on the given information and constraints
• Examples:
  • Determining the feasible region for a linear programming problem (maximizing profit or minimizing cost subject to constraints)
  • Finding the range of possible solutions for a problem with multiple constraints (budget and time limitations)

Interpreting Solutions

• Consider the context of the problem
• Ensure that the answer makes sense and is relevant to the given situation
• Example:
  • A bakery sells cupcakes and cookies. Cupcakes require 2 units of flour and 1 unit of sugar, while cookies require 1 unit of flour and 2 units of sugar. The bakery has 50 units of flour and 60 units of sugar available. If the profit per cupcake is $2 and the profit per cookie is $1, find the number of cupcakes and cookies the bakery should produce to maximize profit.
  • Let x be the number of cupcakes and y be the number of cookies. The constraints are:
    • $2x + y ≤ 50$ (flour constraint)
    • $x + 2y ≤ 60$ (sugar constraint)
    • $x ≥ 0, y ≥ 0$ (non-negativity constraints)
  • The objective function is to maximize profit: $P = 2x + y$
  • Solving the system of inequalities and finding the maximum value of the objective function at the vertices of the feasible region yields the solution: 20 cupcakes and 10 cookies, with a maximum profit of $50.