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Warm Up

Solve Special Types of Linear Systems. Warm Up. Lesson Presentation. Lesson Quiz. (0, –3). ANSWER. 3 specialty pencils. ANSWER. Warm-Up. 1. Solve the linear system. 2 x + 3 y = –9 x – 2 y = 6.

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Warm Up

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  1. Solve Special Types of Linear Systems Warm Up Lesson Presentation Lesson Quiz

  2. (0, –3) ANSWER 3 specialty pencils ANSWER Warm-Up 1.Solve the linear system. 2x + 3y = –9 x – 2y = 6 2. You buy 8 pencils for $8 at the bookstore. Standard pencils cost $0.85 and specialty pencils cost $1.25. How many specialty pencils did you buy?

  3. ANSWER The lines are parallel because they have the same slope but differenty-intercepts. Parallel lines do not intersect, so the system has no solution. Example 1 Show that the linear system has no solution. 3x + 2y = 10 Equation 1 3x + 2y = 2 Equation 2 SOLUTION METHOD 1 Graphing Graph the linear system.

  4. 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. 3x + 2y = 10 3x + 2y = 2 0 = 8 This is a false statement. Example 1 METHOD 2 Elimination Subtract the equations.

  5. ANSWER The equations represent the same line, so any point on the line is a solution. So, the linear system has infinitely many solutions. Graphing METHOD 1 y = x + 2 1 2 Example 2 Show that the linear system has infinitely many solutions. x – 2y = –4 Equation 1 Equation 2 SOLUTION Graph the linear system.

  6. Substitute x + 2 for yin Equation 1 and solve for x. 1 1 1 –4 Substitute x + 2 for y. x – 2 x + 2 = 2 2 2 –4 x – 2y= ANSWER The variables are eliminated and you are left with a statement that is true regardless of the values of xand y. This tells you that the system has infinitely many solutions. –4 –4 = Example 2 METHOD 2 Substitution Write Equation 1. Simplify.

  7. ANSWER No solution. Sample answer: When you solve the system you get 0 = 9, which is a false statement. ANSWER Infinitely many solutions. Sample answer: When you solve the system you get 12 = 12, which is a true statement. Guided Practice Tell whether the linear system has no solution or infinitely many solutions. Explain. 1. 5x + 3y = 6 –5x – 3y = 3 2.y =2x – 4 –6x + 3y =–12

  8. ANSWER Because the lines have the same slope and the same y-intercept, the system has infinitely many solutions. Example 3 Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions. a. 5x + y = –2 Equation 1 –10x – 2y = 4 Equation 2 SOLUTION Write Equation 1 in slope- intercept form. y = –5x – 2 Write Equation 2 in slope- intercept form. y =–5x – 2

  9. 3 5 y = –3x + 2 2 y =–3x – ANSWER Because the lines have the same slope but different y-intercepts, the system has no solution. Example 3 b. 6x + 2y = 3 Equation 1 6x + 2y = –5 Equation 2 SOLUTION Write Equation 1 in slope-intercept form. Write Equation 2 in slope-intercept form.

  10. 3.Without solving the linear system, tell whether it has one solution, no solution, or infinitely many solutions. ANSWER one solution Guided Practice x – 3y = –15 Equation 1 2x – 3y =–18 Equation 2

  11. Example 4 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.

  12. Example 4 SOLUTION 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

  13. –45x – 30y = –465 ANSWER There are infinitely many solutions, so you cannot determine the cost of one glossy print. You need more information. Example 4 STEP2 Solve the linear system using elimination. 45x + 30y = 465 45x + 30y = 465 15x + 10y = 155 0= 0

  14. ANSWER A glossy print costs $8. Guided Practice 4.WHAT IF? In Example 4, suppose a glossy print costs $3 more than a regular print. Find the cost of a glossy print.

  15. 1. 4x + 2y = 12 y = –2x + 8 ANSWER no solution 2 5 2. –2x + 5y = 5 y = x + 1 ANSWER infinitely many solutions Lesson Quiz Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions.

  16. 3. A group of 12 students and 3 teachers pays $57 for admission to a primate research center. Another group of 14 students and 4 teachers pays $69. Find the cost of one student ticket. ANSWER $3.50 Lesson Quiz

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