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Warm Up Evaluate each expression for x = 1 and y =–3. 1. x – 4y –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

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Objectives Identify solutions of linear equations in two variables. Solve systems of linear equations in two variables by graphing.

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**Vocabulary systems of linear equations**

solution of a system of linear equations

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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.

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**Example 1A: Identifying Solutions of Systems**

Tell whether the ordered pair is a solution of the given system. (5, 2); 3x – y = 13 2 – 2 0 0 0 3(5) – 15 – 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.

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**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 –2 + 3(2) 4 x + 3y = 4 – 4 4 –x + y = 2 –(–2) 4 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.

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**All solutions of a linear equation are on its graph**

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. y = 2x – 1 y = –x + 5 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.

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**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). Check Substitute (–1, –1) into the system. y = x y = –2x – 3 (–1) –2(–1) –3 – – 3 –1 – 1 y = x (–1) (–1) –1 –1 • (–1, –1) y = –2x – 3 The solution is (–1, –1).

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** 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. y = x + 5 y = –2x – 1 y = –2x – 1 3 –2(–2) – 1 – 1 y = x + 5 3 –2 + 5 3 3 The solution is (–2, 3).

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**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?

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Example 3 Continued 1 Make a Plan 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. Total pages is number read every night plus already read. Wren y = 2 x + 14 Jenni y = 3 x + 6

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**Example 3 Continued 2 Solve**

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. (8, 30) Nights

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** Example 3 Continued Look Back 3**

Check (8, 30) using both equations. Number of days for Wren to read 30 pages. 2(8) + 14 = = 30 Number of days for Jenni to read 30 pages. 3(8) + 6 = = 30

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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?

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**Check It Out! Example 3 Continued**

1 Make a Plan 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. Total cost is price for each rental plus member- ship fee. Club A y = 3 x + 10 Club B y = 2 x + 15

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**Check It Out! Example 3 Continued**

Solve 2 Graph y = 3x + 10 and y = 2x 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.

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**Check It Out! Example 3 Continued**

Look Back 3 Check (5, 25) using both equations. Number of movie rentals for Club A to reach $25: 3(5) + 10 = = 25 Number of movie rentals for Club B to reach $25: 2(5) + 15 = = 25

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Lesson Quiz: Part I Tell whether the ordered pair is a solution of the given system. 1. (–3, 1); 2. (2, –4); no yes

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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

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**Make a Graphic Organizer**

Must fold Must have color Must write out the problem and include somewhere in the organizer Must show all 3 methods of solving the problem in the organizer (Hint: All 3 answers should be the same!) Be Creative!

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