Systems of Linear Equations and Inequalities

Solving Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations that must be true at the same time.
• The solution to a system of linear equations is the point of intersection of the lines, which is the point that satisfies both equations.
• Methods for solving systems of linear equations:
  • Substitution Method:
    • Solve one equation for one variable.
    • Substitute the expression into the other equation.
    • Solve for the other variable.
  • Elimination Method:
    • Multiply both equations by necessary multiples such that the coefficients of one variable are the same.
    • Add or subtract the equations to eliminate one variable.
    • Solve for the other variable.
  • Graphing Method:
    • Graph both equations on the same coordinate plane.
    • The point of intersection is the solution.

Solving Linear Inequalities

• A linear inequality is an inequality in which the variables are raised to the first power.
• The solution to a linear inequality is a range of values that satisfy the inequality.
• Methods for solving linear inequalities:
  • Graphing Method:
    • Graph the related equation (replace the inequality sign with an equal sign).
    • Identify the boundary point (the point at which the inequality becomes an equality).
    • Shade the region that satisfies the inequality.
  • Algebraic Method:
    • Add or subtract the same value to both sides of the inequality.
    • Multiply both sides of the inequality by a positive number.
    • Flip the inequality sign when multiplying both sides by a negative number.
  • Interval Notation:
    • Write the solution in interval notation (e.g., (-∞, 3] or [2, ∞)).
    • Use parentheses to indicate that the endpoint is not included.
    • Use brackets to indicate that the endpoint is included.

Key Concepts

• The inequality sign can be reversed when multiplying both sides of the inequality by a negative number.
• The solution to a system of linear equations can be a single point, no solution, or infinitely many solutions.
• The solution to a linear inequality can be expressed in interval notation, graphically, or in set-builder notation.