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6.5

6.5. Solve Special Types of Linear Systems. WORDS of the Day !. Perpendicular Lines-. Two lines that intersect to form a right angle. ** Slopes are opposite reciprocals**. Parallel Lines -. Two lines that never intersect and are on the same plane. ** Slopes are the same**. Perpendicular.

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6.5

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  1. 6.5 • Solve Special Types of Linear Systems

  2. WORDS of the Day ! Perpendicular Lines- Two lines that intersect to form a right angle ** Slopes are opposite reciprocals** Parallel Lines - Two lines that never intersect and are on the same plane ** Slopes are the same**

  3. Perpendicular Parallel

  4. Objective Identify the number of solutions of a linear system

  5. Vocabulary • Inconsistent System-A linear system with no solutions • Consistent dependent System- A linear system with infinitely many solutions

  6. Example 1- A linear system with no solutions Show that the linear system has no solution Equation 1 3x + 2y = 2 Equation 2 3x + 2y = 10 Method 1- Graphing Graph the linear system. 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.

  7. Equation 1 3x + 2y = 2 Equation 2 3x + 2y = 10 Method 2- Elimination 2 3x + 2y = 2 - - 10 3x + 2y = 10 Subtract the equations 0 = -8 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.

  8. Example 2- A linear system with infinitely many solutions Show that the linear system has infinitely many solutions Equation 1 y = 1/2 x + 2 Equation 2 x - 2y = -4 Method 1- Graphing Graph the linear system. The equations represent the same line, so any point on the line is a solution. So the linear system has infinitely many solutions.

  9. Equation 2 Equation 1 y = 1/2 x + 2 x - 2y = - 4 Method 2- Substitution Write Equation 2. x - 2y = - 4 x - 2(1/2 x + 2) = - 4 Substitute 1/2 x + 2 for y x - x - 4 = - 4 Distributive Property 0 - 4 = - 4 Simplify - 4 = - 4 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

  10. ANSWER 2. ANSWER 2y = 8x + 4 No Solution -4x + y = 4 Infinite Solutions Check Point!!!!! Solve the linear system using elimination. 1. y = 2x -7 4x - 2y = 14

  11. Number of Solutions of a Linear System Infinitely Many Solutions One Solution No Solution The lines are parallel. The lines have the same slope and different y-intercepts. The lines coincide. The lines have the same slope and the same y-intercepts. The lines intersect. The lines have different slopes

  12. Example 3- Write and solve a system of linear equations ART An artist wants to sell prints of her paintings. She orders a set of prints for each of two of her paintings. Each set contains regular prints and glossy prints, as shown in the table. Find the cost of one glossy print.

  13. –45x – 30y = –465 STEP1 Write a linear system. Let xbe the cost (in dollars) of a regular print, and let ybe the cost (in dollars) of a glossy print. 45x + 30y = 465 Cost of prints for one painting 15x + 10y = 155 Cost of prints for other painting STEP2 Solve the linear system using elimination. 45x + 30y = 465 45x + 30y = 465 15x + 10y = 155 0= 0 There are infinitely many solutions, so you cannot determine the cost of one glossy print. You need more information.

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