WarmUp Exercises 1 Evaluate 3 x 5 y

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Warm-Up Exercises 1. Evaluate – 3 x – 5 y for x = –

Warm-Up Exercises 1. Evaluate – 3 x – 5 y for x = – 3 and y = 4. ANSWER – 11 2. Solve the system by graphing. x +y = 2 2 x + y = 3 ANSWER ( 1, 1 ) 3. Twice a number x plus a number y is 3. The number y subtracted from three times the number x is 7. Find x and y by graphing. ANSWER ( 2, – 1 )

Example 1 Use Substitution Solve the system using substitution. y = 2 x 4

Example 1 Use Substitution Solve the system using substitution. y = 2 x 4 x – y = 6 Equation 1 Equation 2 SOLUTION Substitute 2 x for y in Equation 2. Solve for x. 4 x – y = 6 4 x – 2 x = 6 x=3 Write Equation 2. Substitute 2 x for y. Combine like terms. Solve for x.

Example 1 Use Substitution Substitute 3 for x in Equation 1. Solve for y.

Example 1 Use Substitution Substitute 3 for x in Equation 1. Solve for y. y = 2 x Write Equation 1. y = 2( 3) Substitute 3 for x. y=6 Solve for y. You can check your answer by substituting 3 for x and 6 for y in both equations. ANSWER The solution is ( 3, 6).

Example 2 Use Substitution Solve the system using substitution. 3 x + 2 y

Example 2 Use Substitution Solve the system using substitution. 3 x + 2 y = 7 x – 2 y = – 3 Equation 1 Equation 2 SOLUTION STEP 1 Solve Equation 2 for x. x – 2 y = – 3 x = 2 y – 3 Choose Equation 2 because the coefficient of x is 1. Solve for x to get revised Equation 2.

Example 2 Use Substitution STEP 2 Substitute 2 y – 3 for x in

Example 2 Use Substitution STEP 2 Substitute 2 y – 3 for x in Equation 1. Solve for y. 3 x + 2 y = 7 3( 2 y – 3) + 2 y = 7 6 y – 9 + 2 y = 7 8 y – 9 = 7 8 y = 16 y=2 Write Equation 1. Substitute 2 y – 3 for x. Use the distributive property. Combine like terms. Add 9 to each side. Solve for y.

Example 2 Use Substitution STEP 3 Substitute 2 for y in revised Equation 2.

Example 2 Use Substitution STEP 3 Substitute 2 for y in revised Equation 2. Solve for x. x = 2 y – 3 Write revised Equation 2. x = 2( 2 ) – 3 Substitute 2 for y. x=1 Simplify. STEP 4 Check by substituting 1 for x and 2 for y in the original equations. Equation 1 3 x + 2 y = 7 Equation 2 Write original equations. x – 2 y = – 3

Example 2 Use Substitution ? 3( 1 ) + 2( 2 ) = 7

Example 2 Use Substitution ? 3( 1 ) + 2( 2 ) = 7 ? 3+4=7 7=7 ANSWER The solution is ( 1, 2). Substitute for x and y. Simplify. Solution checks. ? 1 – 2( 2 ) = – 3 ? 1 – 4 = – 3

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 1. 2 x + y = 3 3 x + y = 0 ANSWER (– 3, 9 ). Sample answer: The second equation; this equation had 0 on one side and the coefficient of y was 1, so I solved for y to obtain y = – 3 x.

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 2. 2 x + 3 y = 4 x + 2 y = 1 ANSWER ( 5, – 2). Sample answer: The second equation; the coefficient of x in this equation was 1, so solving for x gave a result that did not involve any fractions.

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to

Checkpoint Use Substitution Solve the system using substitution. Tell which equation you chose to solve and use for the substitution. Explain. 3. 3 x – y = 5 4 x + 2 y = 10 ANSWER ( 2, 1). Sample answer: The first equation; the coefficient of y in this equation was – 1, so solving for y gave a result that did not involve any fractions.

MUSEUM CURATOR Curators are responsible for obtaining artifacts and specimens for display at museums,

MUSEUM CURATOR Curators are responsible for obtaining artifacts and specimens for display at museums, zoos, nature centers, or historic sites. They also study, document, and help preserve the items they acquire.

Example 3 Write and Use a Linear System Museum Admissions On one day, the

Example 3 Write and Use a Linear System Museum Admissions On one day, the Henry Ford Museum in Dearborn, Michigan, admitted 4400 adults and students and collected $57, 200 in ticket sales. The price of admission is $14 for an adult and $10 for a student. How many adults and how many students were admitted to the museum that day? SOLUTION VERBAL MODEL Number of adults + Number of students = Total number admitted Adult Number Student Number Total amount • • = + price of adults price of students collected

Example 3 LABELS ALGEBRAIC MODEL Write and Use a Linear System Number of adults

Example 3 LABELS ALGEBRAIC MODEL Write and Use a Linear System Number of adults = x (adults) Number of students = y (students) Total number admitted = 4400 (people) Price for one adult = 14 (dollars) Price for one student = 10 (dollars) Total amount collected = 57, 200 (dollars) x + y = 4400 Equation 1 (number admitted) 14 x + 10 y = 57, 200 Equation 2 (amount collected)

Example 3 Write and Use a Linear System Use substitution to solve the linear

Example 3 Write and Use a Linear System Use substitution to solve the linear system. x = 4400 – y Solve Equation 1 for x; revised Equation 1. 14 (4400 – y) + 10 y = 57, 200 Substitute 4400 – y for x in Equation 2. 61, 600 – 14 y + 10 y = 57, 200 Use the distributive property. 61, 600 – 4 y = 57, 200 – 4 y = – 4400 y = 1100 Combine like terms. Subtract 61, 600 from each side. Divide each side by – 4.

Example 3 Write and Use a Linear System x = 4400 – y Write

Example 3 Write and Use a Linear System x = 4400 – y Write revised Equation 1. x = 4400 – 1100 Substitute 1100 for y. x = 3300 Simplify. ANSWER There were 3300 adults and 1100 students admitted to the Henry Ford Museum that day.

Checkpoint Write and Use a Linear System 4. On another day, the Henry Ford

Checkpoint Write and Use a Linear System 4. On another day, the Henry Ford Museum admitted 1300 more adults than students and collected $56, 000. How many adults and how many students were admitted to the museum that day? ANSWER 2875 adults and 1575 students

Checkpoint Write and Use a Linear System 5. Solve the system of equations in

Checkpoint Write and Use a Linear System 5. Solve the system of equations in Example 3 by solving Equation 1 for y instead of x. Compare your solution to the solution in Example 3. What conclusion can you make? ANSWER The answers are the same; you can solve for either variable to solve a system.

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