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7.5 Solve Special Types of Linear Systems You will identify systems that have no solutions and systems that have infinitely many solutions. Essential Question: How can you identify the number of solutions in a linear system? Warm-Up Exercises Lesson 7.5, For use with pages 459-465 1. Solve the linear system. 2x + 3y = – 9 x – 2y = 6 ANSWER (0, – 3) 2. You buy 8 pencils for $8 at the bookstore. Standard pencils cost $.85 and specialty pencils cost $1.25. How many specialty pencils did you buy? ANSWER 3 specialty pencils Independent Practice Check Textbook p. 447-450 Independent Practice Check Textbook p. 454-457 Vocabulary Exactly One Solution = Consistent and Independent Graphing – Intersecting Lines Substitution – Produces a Solution Elimination – Produces a Solution Infinite Solutions = Consistent and Dependent Graphing – Same Line Substitution – Variables all disappear – Left with True Sentence Elimination – Variables all disappear – Left with True Sentence No solutions = Inconsistent Graphing - Parallel Lines Substitution – Variables all disappear – Left with False Sentence Elimination – Variables all disappear – Left with False Sentence 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 SOLUTION METHOD 1 Graphing Graph the linear system. 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 METHOD 2 Elimination Subtract the equation. 3x + 2y = 10 3x + 2y = 2 0 = 8 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. EXAMPLE 2 A linear system with infinitely many solutions Show that the linear system has infinitely many solutions. x – 2y = – 4 Equation 1 y = 1x + 2 Equation 2 2 SOLUTION METHOD 1 Graphing 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 Substitute 1 x + 2 for y in Equation 1 and solve for x. 2 x – 2y = – 4 Write Equation 1 x – 2 1x + 2 = – 4 Substitute 1 x + 2 for y. 2 2 –4= –4 Simplify. ANSWER The variables are eliminated and you are left with a true statement regardless of the values of x and y. This tells you that the system has infinitely many solutions. GUIDED PRACTICE for Examples 1 and 2 Tell whether the linear system has no solution or infinitely many solutions. Explain. 1. 5x + 3y = 6 Equation 1 – 5x – 3y = 3 Equation 2 METHOD 2 Elimination Subtract the equations. 5x + 3y = 6 – 5x – 3y = 3 0=9 This is a false statement. GUIDED PRACTICE for Examples 1 and 2 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. GUIDED PRACTICE for Examples 1 and 2 2. y = 2x – 4 Equation 1 – 6x + 3y = – 12 Equation 2 METHOD 2 Substitution Substitute 2x – 4 for y in Equation 2 and solve for x. – 6x + 3y = – 12 Write Equation 2 – 6x + 3(2x – 4) = – 12 Substitute (2x – 4) for y. – 12 = – 12 Simplify. GUIDED PRACTICE for Examples 1 and 2 ANSWER The variables are eliminated and you are left with a true statement regardless of the values of x and y. This tells you that the system has infinitely many solution. EXAMPLE 3 Identify the number of solutions. 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- y = – 5x – 2 intercept form. Write Equation 2 in slope- y = – 5x – 2 intercept form. ANSWER Because the lines have the same slope and the same y-intercepts,the system has infinitely many solutions. They are the same line. EXAMPLE 3 Identify the number of solutions. b. 6x + 2 y = 3 Equation 1 6x + 2 y = – 5 Equation 2 SOLUTION b. y = – 3x + 3 Write Equation 1 in slope- intercept form. 2 y = – 3x – 5 Write Equation 2 in slope-intercept form. 2 ANSWER Because the lines have the same slope but different y- intercepts, the system has no solution. They are parallel lines. EXAMPLE 4 Write and solve a system of linear equations ART An artist wants to sell prints Regular Glossy Cost of her paintings. She orders 45 30 $465 a set of prints for each of two of her paintings. Each 15 10 $155 set contains regular prints and glossy prints, as shown in the table. Find the cost of one glossy print. EXAMPLE 4 Write and solve a system of linear equations SOLUTION STEP 1 Write a linear system. Let x be the cost (in dollars) of a regular print, and let y be 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 EXAMPLE 4 Write and solve a system of linear equations STEP 2 Solve the linear system using elimination. 45x + 30y = 465 45x + 30y = 465 15x + 10y = 155 – 45x – 30y = – 465 0=0 ANSWER There are infinitely many solutions, so you cannot determine the cost of one glossy print. You need more information. GUIDED PRACTICE for Examples 3 and 4 3. Without solving the linear system, tell whether it has one solution, no solution, or infinitely many solutions. one solution. x – 3y = – 15 Equation 1 2x – 3y = – 18 Equation 2 SOLUTION x+5 Write Equation 1 in slope- y= intercept form. 3 2x + 18 Write Equation 2 in slope- y= intercept form. 3 ANSWER Because the lines have different slope and different y-intercepts,the system has one solution. Essential Question: How can you identify the number of solutions in a linear system? Independent Practice Textbook p. 462-465