Solving Linear Systems by Substitution Objective: Students will solve a linear system by substitution method. Use this method when ONE of the equation has been solved for a variable.

Algebra Standards: 8 EE 8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Review… Solve the system by graphing x = 3 y = 2x - 1 Solution (3, 5)   x = 3

Review… Solve for y when x = 3 Solution (3, 5) So, when x = 3… Substitute the 3 into the x Solve for y when x = 3   So, when x = 3… y = 5 based on the rule ( y = 2x -1)     Solution (3, 5)  

{ 5 5 Solve the linear system y = 4x Check answer! x + y = 10 #1 Substitution Solve the linear system { y = 4x Check answer! x + y = 10 Solution: y = 4x x + y = 10 x + 4x = 10 1 y = 4 (2) 5x = 10 5 5 y = 8 x = 2 Solution is (2, 8)

{ 5 5 Solve the linear system x = 3y Check answer! 2x – y = 10 #2 Substitution Solve the linear system { x = 3y Check answer! 2x – y = 10 Solution: x = 3y 2x – y = 10 2 (3y) – y = 10 x = 3 (2) 6y – y = 10 1 5y = 10 x = 6 5 5 y = 2 Solution is (6, 2)

{ x = -3y – 9 x = -3 (-4) – 9 x = 12 – 9 x = 3 -11y = 44 -11 -11 #3 Substitution Solve the linear system { x + 3y = -9 Check answer! 2x – 5y = 26 Solution: x + 3y = -9 2x – 5y = 26 -3y -3y 2 (-3y – 9) – 5y = 26 x = -3y – 9 -6y – 18 – 5y = 26 x = -3 (-4) – 9 -11y – 18 = 26 x = 12 – 9 +18 +18 x = 3 1 -11y = 44 Solution is (3, -4) -11 -11 y = -4

{ +x +x y = x +1 y = -1 + 1 -1 -1 y = 3x = -3 3 3 #4 Substitution Solve the linear system { -x + y = 1 Check answer! 2x + y = -2 Solution: -x + y = 1 2x + y = -2 +x +x 2x + x + 1 = -2 y = x +1 3x + 1 = -2 y = -1 + 1 -1 -1 y = 1 3x = -3 3 3 Solution is (-1, 0) x = -1

#5b Substitution- Word Problem In one day the National Civil Rights Museum in Memphis, Tennessee, admitted 321 adults and children and collected $1590. The price of admission is $6 for an adult and $4 for a child. How many adults and how many children were admitted to the museum that day? Solution: How many x = adults Money y = children 6x + 4y x + y = 321 = 1590 -y -y 6(-y + 321) + 4y = 1590 x = -y + 321 -6y + 1926 + 4y = 1590 -2y + 1926 = 1590 x = -168 + 321 -1926 -1926 x = 153 153 adults -2y -336 = 168 children y = 168

#6 Substitution The length of a rectangle is 5 centimeters more than three times the width. If the perimeter of the rectangle is 34 centimeters, what are its dimensions? P = L + L + w +w { L = 3w + 5 P = 2L + 2w 34 = 2L + 2w Check answer! Solution: 2L + 2w = 34 L = 3w + 5 2 (3w + 5) + 2w = 34 L = 3 (3) + 5 6w + 10 + 2w = 34 8w + 10 = 34 L = 9 + 5 -10 -10 L = 14 1 8w = 24 w = 3 Length is 14 cm and the width is 3cm 8 8

-9 -9 3 3 The sum of the ages of Petra and her mother is 54. Her #7 Substitution The sum of the ages of Petra and her mother is 54. Her mother is 9 years more than twice as old as Petra. How old are Petra and her mother Check answer! Solution: x = Petra’s age x + y = 54 y = 2x + 9 y = mother’s age x + 2x + 9 = 54 y = 2( ) 15 + 9 3x + 9 = 54 y = 30 + 9 -9 -9 1 3x = 45 y = 39 3 3 x = 15 Petra is 15 years old. Her mother is 39 years old.

Assignment Book Pg. 284 # 5, 11, 13, 14, 17