1. Objective I will identify the number of solutions a linear system has using one of the three methods used for solving linear systems.

2. If the graphs of the equations cross once, there will be one solution. If the graphs of the equations are parallel, there will be no solutions. If the graphs of the equations are the same line, there will be an infinite number of solutions. Number of Solutions

3. Method 1: Function Form Finding number of solutions Put equations in function form (isolate y; y = mx + b) Different slopes (m) - one intersection; one solution Same slope (m), different y-intercept (b) - parallel lines, no solution Same slope (m) and y-intercept (b) - same line, infinite solutions

4. Method 2: Substitution Finding number of solutions Solve for one variable in one equation Substitute into other equation Equation is true - one solution Equation is false - no solution 0 = 0 - infinite solutions

5. Method 3: Linear Combinations Finding number of solutions Multiply equations (if necessary) so one variable has an opposite Add equations Solve Equation is true - one solution Equation is false - no solution 0 = 0 - infinite solutions

6. Example Find the number of solutions. 2x + y = 5 2x + y = 1 Method 1: Function Form y = -2x + 5 y = -2x + 1 Same slope, different y-intercept - no solution

7. Example Find the number of solutions. 2x + y = 5 2x + y = 1 Method 2: Substitution y = -2x + 5 2x + (-2x + 5) = 1 2x - 2x + 5 = 1 0 + 5 = 1 5 ≠ 1 Equation is false - no solution

8. Example Find the number of solutions. 2x + y = 5 2x + y = 1 Method 3: Linear Combinations 2x + y = 5 -2x - y = -1 0 ≠ 4 Equation is false - no solution

9. Guided Practice Find the number of solutions. 1. 8x + 4y = 16 3y = 6x - 15 2. -2y = -2x + 4 5y = 5x - 10

10. Independent Practice Find the number of solutions. 1. 6y = 12x + 18 -2x + y = 18 2. y = 14x - 3 -42x - 3y = -9