Betty Bob has six more nickels than dimes. The total amount of money she has is $3.30. How many of each coins does she have? Warm Up

SOLUTION EXAMPLE 1 A linear system with no solution Show that the linear system has no solution. 3x + 2y = 10 Equation 1 3x + 2y = 2 Equation 2 Graph the linear system. METHOD 1 Graphing

EXAMPLE 1 A linear system with no solution ANSWER The lines are parallel because they have the same slope but different y- intercepts. Parallel lines do not intersect, so the system has NO SOLUTION.

EXAMPLE 1 A linear system with no solution ANSWER The variables are eliminated and you are left with a false statement regardless of the values of x and y. This tells you that the system has NO SOLUTION. Multiply one equation by -1 and add the equations. METHOD 2 Elimination 3x + 2y = 10 3x + 2y = 2 This is a false statement. -3x - 2y = -10 3x + 2y = 2 0 = -8

Graph the linear system to see if there is a solution. TRY THIS

Graph the linear system to see if there is a solution. The lines are parallel because they have the same slope but different y- intercepts. Parallel lines do not intersect, so the system has NO SOLUTION.

Use Elimination or Substitution to see if there is a solution. This is a false statement. ANSWER The variables are eliminated and you are left with a false statement regardless of the values of x and y. This tells you that the system has NO SOLUTION. Multiply one equation by -1 and add the equations. YOU TRY

EXAMPLE 2 A linear system with infinitely many solutions Show that the linear system has infinitely many solutions. Equation 1 Equation 2 SOLUTION Graphing METHOD 1 Graph the linear system.

EXAMPLE 2 A linear system with infinitely many solutions ANSWER The equations represent the same line, so any point on the line is a solution. So, the linear system has INFINITELY MANY SOLUTIONS.

EXAMPLE 2 A linear system with infinitely many solutions METHOD 2 Substitution The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This tells you that the system has INFINITELY MANY SOLUTIONS. ANSWER Equation 1 Equation 2 Substitute Eq 2 into Eq 1 This is a true statement

Show that the linear system has infinitely many Solutions using the graphing method. Equation 1 Equation 2 SOLUTION Graphing METHOD 1 Graph the linear system. YOU TRY

Show that the linear system has infinitely many solutions. ANSWER The equations represent the same line, so any point on the line is a solution. So, the linear system has INFINITELY MANY SOLUTIONS.

METHOD 2 Substitution The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This tells you that the system has INFINITELY MANY SOLUTIONS. ANSWER Equation 1 Equation 2 Substitute Eq 2 into Eq 1 This is a true statement YOU TRY

Some of the following practice problems may have one solution, no solution or infinitely many solutions. Page 463 # Use Graphing #18 – 20 Use Elimination or Substitution

Exit Ticket This system may have one solution, no solution or infinitely many solutions. Use Elimination or Substitution