1. Solving Systems of Linear Equations in Two Variables: When you have two equations, each with x and y, and you figure out one value for x and one value for y that will make BOTH equations true.

2. Three methods of solving systems: Graphing Method Substitution Method Elimination Method

3. Graphing Method Solve each equation for y, then graph each equation on a rectangular coordinate plane, then observe where the two lines intersect. The coordinates of the intersection point are the solutions of the system. Benefit: very easy to do Drawback :requires graph paper might be imprecise

4. Example: Solve by graphing : 2x + 3y = 7 - x + 2y = 14 (Go to Sketchpad sketch)

6. Three methods of solving systems: Graphing Method Substitution Method Elimination Method

7. In both of the next two methods, the Substitution Method and the Elimination Method, the process will involve the sub- goal of “getting rid of” one of the two variables, in order to form an equivalent equation in ONE variable (remember, it is easy to solve a linear equation in one variable!). After you solve for the one variable, you then plug the solution back into either of the original equations, and solve for the other variable.

8. Substitution Method In one of the equations, solve for one of the variables, either “ y = some expression with x” or “ x = some expression with y”. Then, substitute that new expression into the other equation, thereby getting rid of one of the variables. Then, simply solve the linear equation for the remaining variable.

9. Example: Solve by substitution : 2x + 3y = 7 and – x + 2y = 14

11. Elimination Method Multiply both sides of one (or both) equation(s) by the same number(s) in order to get the coefficients of one of the variables in each equation to be OPPOSITES (like 12y and -12y). Then, add the two equations to form a new equivalent expression in ONE variable, since the other variable has been “eliminated”. Now, solve the linear equation for the remaining variable.

12. Example: Solve by elimination : 2x + 3y = 7 and – x + 2y = 14

13. When does each method work best? Graphing Method good if you have graph paper AND if your solution does not have to be precise Substitution Method good if one of the variables has a coefficient of (invisible) 1, like 6x + y = 15 Elimination Method good if the coefficients of one of the variables divides evenly into the same variable’s coefficient in the other equation, like 3x + 5y = 9, and -2x + 10y = -8