MATH 1100 SECTION 7.1 Notes Systems of Equations – Text Pages 447-453 Systems of Equations: When solving a problem with more than one unknown, you must have more than one equation. This set of equations is called a System of Equations. Note: The number of equations should equal the number of unknowns. Solution to a System of Equations: A solution to a system of equations in two variables is an ordered pair of numbers that makes both equations true. Solving Systems of Equations: There are three methods for solving systems of equations. method is graphical, the other two methods are algebraic. One Graphical Method: If the system of equations being solved contains equations that can be easily graphed on a calculator, or are graphs that you can easily sketch (like linear equations), then we can solve the system by finding the point(s) at which all graphs in the system meet. This method gives an estimate to the solution and is not always easy to do, so we will concentrate on the algebraic methods of solving systems. Substitution Method: To use the substitution method to solve a system of equations, we need to solve one equation for one variable and then substitute that variable into the other equation. Elimination Method: To use the elimination method to solve a system of equations, use the addition principle to cancel terms. Example 1: (Substitution Method) Solve the following system of equations using the substitution method. 2 x y 7 3x y 13 solve one equation for a variable: substitute into the other equation and solve for the other variable: substitute solution into the initial equation and solve for the remaining solution: Example 2: (Substitution Method) Solve the following system of equations using the substitution method. x2 y 9 x y 3 0 solve one equation for a variable: substitute into the other equation and solve for the other variable: substitute solution into the initial equation and solve for the remaining solution: Example 3: (Elimination Method) Solve the following system of equations using the elimination method. 4 x 3 y 10 9 x 4 y 1 multiply entire equations by appropriate values and add the two equations together: solve for the remaining variable: plug solution into one of the original equations and solve for other variable: check and state: Example 4: (Elimination Method) Solve the following system of equations using the elimination method. 3 x 2 4 y 17 2 2 x 5 y 2 multiply entire equations by appropriate values and add the two equations together: solve for the remaining variable: plug solution into one of the original equations and solve for other variable: check and state: Example 5: Solve the following system of equations with the elimination method and with the substitution method. x y 180 x 2y 5 Substitution Method: Elimination Method: Example 6: Find the lengths of the sides of a right triangle if the length of the hypotenuse is 25 ft and the area is 84 ft2.