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## Solving Systems by Graphing

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**Solving Systems by Graphing**Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1**Warm Up**Evaluate each expression for x = 1 and y =–3. 1.x – 4y2. –2x + y Write each expression in slope-intercept form. 3.y –x = 1 4. 2x + 3y =6 5. 0 = 5y + 5x 13 –5 y = x + 1 y =x + 2 y = –x**Objectives**Identify solutions of linear equations in two variables. Solve systems of linear equations in two variables by graphing.**Vocabulary**systems of linear equations solution of a system of linear equations**A system of linear equations is a set of two or more linear**equations containing two or more variables. A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.**3x – y =13**3(5) – 2 13 0 2 – 2 0 15 – 2 13 0 0 13 13 Example 1A: Identifying Solutions of Systems Tell whether the ordered pair is a solution of the given system. (5, 2); 3x – y = 13 Substitute 5 for x and 2 for y in each equation in the system. The ordered pair (5, 2) makes both equations true. (5, 2) is the solution of the system.**Helpful Hint**If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations.**–x + y = 2**x + 3y = 4 –(–2) + 2 2 –2 + 3(2) 4 4 2 –2 + 6 4 4 4 Example 1B: Identifying Solutions of Systems Tell whether the ordered pair is a solution of the given system. x + 3y = 4 (–2, 2); –x + y = 2 Substitute –2 for x and 2 for y in each equation in the system. The ordered pair (–2, 2) makes one equation true but not the other. (–2, 2) is not a solution of the system.**2x + y = 5**(1, 3); –2x + y = 1 2x + y = 5 –2x + y = 1 2(1) + 3 5 –2(1) + 3 1 –2 + 3 1 2 + 3 5 1 1 5 5 Check It Out! Example 1a Tell whether the ordered pair is a solution of the given system. Substitute 1 for x and 3 for y in each equation in the system. The ordered pair (1, 3) makes both equations true. (1, 3) is the solution of the system.**x– 2y = 4**3x + y = 6 3(2)+(–1) 6 2 – 2(–1) 4 6 – 1 6 2 + 2 4 5 6 4 4 Check It Out! Example 1b Tell whether the ordered pair is a solution of the given system. x –2y = 4 (2, –1); 3x + y = 6 Substitute 2 for x and –1 for y in each equation in the system. The ordered pair (2, –1) makes one equation true, but not the other. (2, –1) is not a solution of the system.**y = 2x – 1**y = –x + 5 All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection. The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.**Helpful Hint**Sometimes it is difficult to tell exactly where the lines cross when you solve by graphing. It is good to confirm your answer by substituting it into both equations.**Check**Substitute (–1, –1) into the system. y = –2x– 3 y = x (–1) (–1) (–1)–2(–1)–3 –12– 3 –1 –1 –1 – 1 Example 2A: Solving a System by Graphing Solve the system by graphing. Check your answer. y = x Graph the system. y = –2x – 3 The solution appears to be at (–1, –1). y = x • (–1, –1) y = –2x – 3 The solution is (–1, –1).**y + x = –1**y + x = –1 y = x –6 −x−x y = Example 2B: Solving a System by Graphing Solve the system by graphing. Check your answer. y = x –6 Graph using a calculator and then use the intercept command. Rewrite the second equation in slope-intercept form.**Check Substitute into the system.**y = x–6 y = x –6 + – 1 – 6 –1 –1 –1 – 1 The solution is . Example 2B Continued Solve the system by graphing. Check your answer.**y = x + 5**y = x+ 5 y = –2x– 1 3–2+ 5 3 –2(–2)– 1 y = –2x – 1 3 3 3 4 – 1 3 3 Check It Out! Example 2a Solve the system by graphing. Check your answer. y = –2x – 1 Graph the system. y = x + 5 The solution appears to be (–2, 3). Check Substitute (–2, 3) into the system. The solution is (–2, 3).**2x + y = 4**2x + y = 4 –2x – 2x y = –2x + 4 Check It Out! Example 2b Solve the system by graphing. Check your answer. Graph using a calculator and then use the intercept command. 2x + y = 4 Rewrite the second equation in slope-intercept form.**2x + y = 4**2x + y = 4 –2(3) – 3 2(3) + (–2) 4 6 – 2 4 4 4 –2 1 – 3 –2 –2 Check It Out! Example 2b Continued Solve the system by graphing. Check your answer. 2x + y = 4 Check Substitute (3, –2) into the system. The solution is (3, –2).**Example 3:Problem-Solving Application**Wren and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?**1**Understand the Problem Example 3 Continued The answer will be the number of nights it takes for the number of pages read to be the same for both girls. List the important information: Wren on page 14 Reads 2 pages a night Jenni on page 6 Reads 3 pages a night**Make a Plan**Total pages every night already read. number read is plus Wren y = 2 x 14 + 2 x y 3 + Jenni = 6 Example 3 Continued Write a system of equations, one equation to represent the number of pages read by each girl. Let x be the number of nights and y be the total pages read.**3**Solve (8, 30) Nights Example 3 Continued Graph y = 2x + 14 and y = 3x + 6. The lines appear to intersect at (8, 30). So, the number of pages read will be the same at 8 nights with a total of 30 pages.****2(8) + 14 = 16 + 14 = 30 3(8) + 6 = 24 + 6 = 30 4 Look Back Example 3 Continued Check (8, 30) using both equations. Number of days for Wren to read 30 pages. Number of days for Jenni to read 30 pages.**Check It Out! Example 3**Video club A charges $10 for membership and $3 per movie rental. Video club B charges $15 for membership and $2 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost?**1**Understand the Problem Check It Out! Example 3 Continued The answer will be the number of movies rented for which the cost will be the same at both clubs. • List the important information: • Rental price: Club A $3 Club B $2 • Membership: Club A $10 Club B $15**Make a Plan**Total cost member- ship fee. for each rental is price plus Club A y = 3 x 10 + 2 x y 2 + 15 Club B = Check It Out! Example 3 Continued Write a system of equations, one equation to represent the cost of Club A and one for Club B. Let x be the number of movies rented and y the total cost.**3**Solve Check It Out! Example 3 Continued Graph y = 3x + 10 and y = 2x + 15. The lines appear to intersect at (5, 25). So, the cost will be the same for 5 rentals and the total cost will be $25.****3(5) + 10 = 15 + 10 = 25 2(5) + 15 = 10 + 15 = 25 4 Look Back Check It Out! Example 3 Continued Check (5, 25) using both equations. Number of movie rentals for Club A to reach $25: Number of movie rentals for Club B to reach $25:**Lesson Quiz: Part I**Tell whether the ordered pair is a solution of the given system. 1. (–3, 1); 2. (2, –4); no yes**Lesson Quiz: Part II**Solve the system by graphing. 3. 4. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be? y + 2x = 9 (2, 5) y = 4x – 3 4 months 13 stamps