Objective

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

•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

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

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

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

Example

Find the number of solutions.2x + y = 52x + y = 1Method 1: Function Formy = -2x + 5y = -2x + 1Same slope, different y-intercept - no

solution

Example Find the number of solutions.2x + y = 52x + y = 1Method 2: Substitutiony = -2x + 52x + (-2x + 5) = 12x - 2x + 5 = 10 + 5 = 15 ≠ 1Equation is false - no solution

Example Find the number of solutions.2x + y = 52x + y = 1Method 3: Linear Combinations2x + y = 5-2x - y = -1 0 ≠ 4

Equation is false - no solution

Guided Practice

Find the number of solutions.1. 8x + 4y = 16 3y = 6x - 152. -2y = -2x + 4 5y = 5x - 10

Independent Practice

Find the number of solutions.1. 6y = 12x + 18 -2x + y = 182. y = 14x - 3

-42x - 3y = -9