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Mastering Systems of Equations: Graphing, Substitution, Elimination, and Linear Programming

This comprehensive guide covers essential methods for solving systems of equations, including graphing, substitution, and elimination. Learn how to approach 3x3 equations and apply linear programming to optimize resource management in business and government contexts. Understand key concepts such as independent, inconsistent, and dependent systems, as well as methods for depicting inequalities graphically. Step-by-step instructions for using matrix functions in graphing calculators are also provided, ensuring you are well-equipped to tackle any system of equations.

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Mastering Systems of Equations: Graphing, Substitution, Elimination, and Linear Programming

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  1. Chapter3 By: elisaguzman & Gavidiaz

  2. What? • Solve systems by graphing • Solve systems by substitution • By elimination • By linear programming • Inequalities by graphing • 3 by 3 equations

  3. Words You Need To Know • System- two or more equations or inequalities • Solution to a system- intersection points • Linear programming- a method used by businesses and government to manage resources and time.

  4. Cheat sheet • One solution- system name: independent, picture name: intersecting • No solution- system name: inconsistent, picture name: parallel • Infinitely Many Solutions- system name: dependent, picture name: consistent

  5. Ways to know • Elimination • Substitution • graphing • 3 by 3 equations

  6. Ways to know • Inequalities by graphing • Linear programming

  7. StepsFor solving a 3 by 3 equation • 1) press 2nd and x-1 to pull up the matrix function • 2) Go over to edit and press enter. To set the matrix for a system of three variables with three equations, the numbers at the top of the screen should read 3x4. Change them if they do not say this. The first number represents the number of variables. The second number represents how many terms there are. • 3) Type the coefficients from your system into the matrix making sure that you enter them alphabetically and that the solution to each equation is the last number on each line.

  8. Steps Continued… • 3) continued… Remember enter 0 if the variable is missing and 1 if there is not a number in front of the variable. • 4) Press 2nd and mode to get to the main screen. Press 2nd and x-1 again. Go over to math and scroll up until your cursor is over “B: rref(“ Press enter to select it. Press 2nd and x-1 and select the first option 1: [A]. Press enter one more time. • 5)Your answers will be the numbers in the last column. BUT watch out for special cases! If the bottom row has all 0’s then the answer is “infinitely many solutions”.

  9. Steps Continued… • 5)…If the bottom row has two 0’s, a 1 and another 0 then the answer is “no solution”.

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