The cost to join an art museum is P600. If you are a member, you can take a lesson at the museum for P20 each. If you are not a member, lessons cost P60 each. Write an equation to find the number of x of lessons after which the total cost y of the lessons with membership is the same as the total cost of lessons without a membership.

Getting Ready! 1. Complete the table below for x + y = 4. X y 2. Complete the table below for 2x – y = 5. X y Can you name a pair that satisfies both equation?

Linear system Consist of two or more linear equations in the same variables x + 2y = 7 and 3x – 2y = 5 Is an ordered pair that satisfies each of the equation in the systems

X y X y Point of intersection

1) -5x + y = 0 and 5x + y = 10 (1, 5) Point of intersection y = 5x 5 = 5(1) 5 = 5 y = -5x = -5(1) = = 5

2) x – y = 5 and 3x + y = 3 (2, -3) Point of intersection y = x – 5 -3 = 2 – 5 -3 = -3 y = -3x = -3(2) = = -3

It has at least one solution. It has different graphs.

Inconsistent system of linear equations – does not have a solution. Dependent system of linear equations – with identical graph.

Steps in Solving for Systems by Graphing

3) x + y = 4 and 2x – y = 5 (3, 1) Point of intersection y = -x = -(3) = 1 y = 2x = 2(3) – 5 1 = = 1

4) x – y = 1 and x + y = 3 (2, 1) Point of intersection y = x – 1 1 = 2 – 1 1 = 1 y = -x = -(2) = = 1

Homework Solve the following linear systems by graphing. 1. y = -x + 3 and y = x x + y = 15 and y = -15

1. y = -x + 3 y = x x + y = 15 y = -15

A gardening company placed orders with a nursery. One was for 13 bushes and 4trees, and totaled P487. The second order was for 6 bushes and 2 trees, and totaled P232. The bill doesn't tell the amount of per item. What were the costs of one bush and of one tree?

Getting Ready! Simplify the following: 1.3(2x + 3) 2.-4(3x – 4) Can you solve the system of linear equation by substitution? Substitute x – 3 for y and simplify the following: 1.5y 2.3y (y+3) Solve for x and y 1. y = 2x + 1 and 3x + 2y = 9

y=2x + 1 and 3x + 2y = 9 Eq. 1 is already solved for y. 3x + 2y = 9eq. 2 3x + 2(2x + 1) = 9 3x + 4x + 2 = 9 7x = 9-2 7x = 7 x = 1 y = 2x + 1eq. 1 y = 2(1) + 1 y = y = 3

a + b = 7 and 3a + 2b = 16 3a + 2b = 16eq. 2 3a + 2(-a + 7) = 16 3a – 2a + 14 = 16 a = a = 2 b = -a + 7eq. 1 b = -(2) + 7 b = b = 5 a + b = 7eq. 1 b = -a + 7

Steps in Solving for Systems by Substitution

x – 2y = -6 and 4x + 6y = 4 4x + 6y = 4eq. 2 4(2y – 6) + 6y = 4 8y – y = 4 14y = y = 28 y = 2 x = 2y - 6eq. 1 x = 2(2) - 6 x = x = -2 x – 2y = -6 x = 2y - 6

m = 2n + 5 and 3n + m = 10 3n + m = 10eq. 2 3n + (2n + 5) = 10 3n + 2n + 5 = 10 5n = n = 1 m = 2n + 5eq. 1 m = 2(1) + 5 m = m = 7 Eq. 1 is already solved for m. 5n = 5

Homework Solve the following linear systems by substitution 1. x = y + 3 and 2x – y = a – 7b = -14 and a- 2b =-4

Seatwork

Solve for the following: (3x -2y) + (3x + 3y) = (3x - 4y) + (-3x + 2y) = (x + y) - (x - 4y) (-2x - 4y) + (-4x + 4y)

Developing Skills Add the following equation. Which variable is eliminated? x + 2y = 5 3x – 2y = -1 y Solve for the remaining variable. x + 2y = y = 5 2y = 5-1 2y = 4 y = 2

Add the following equation. Which variable is eliminated? 2a – 3b = 14 a + 3b = -2 b Solve for the remaining variable. a + 3b = b = -2 3b = b = -6 b = -2

Developing Skills Add the following equation. Which variable is eliminated? 2x + 3y = 11 2x - 5y = 13 x Solve for the remaining variable. 2x + 3y = 11 2x + 3(3) = 11 2x = x = 2 x = 1

Steps in Solving for Systems by adding and subtracting

Add the following equation. Which variable is eliminated? 2a – 3b = 26 -2a - 3b = -2 b Solve for the remaining variable. 2a - 3b = 26 2a – 3(-4) = 26 2a = a = 24 a = 12

5x + 2y = 16 3x – 4y = 20 Can we eliminate a variable by adding and subtracting? (5x + 2y = 16)2 10x + 4y = 32 3x – 4y = 20

10x + 4y = 32 3x – 4y = 20 3x – 4y = 20 3(4) – 4y = 20 – 4y = – 4y = 8 y = -2

6x + 5y = 19 -6x - 9y = -15 6x + 5y = 19 6x +5(-1) = 19 6x = x = 24 x = 4 6x + 5y = 19 2x + 3y = 5 ( )-3

8x + 10y = x+10y = -45 4x + 5y = 35 4(5) +5y = 35 5y = y = 15 y = 3 4x + 5y = 35 -3x + 2y = -9 ( ) 5 ( ) 2

Seatwork

Homework Solve the following linear systems by Elimination 1. 3x - 7y = 5 and 9y= 5x a + 2b = 4 and 2b =8 – 5a

Short Quiz Solve the following linear systems by Elimination 1. x + 4y = 22 and 4x – y = x – 3y = 10 and x + 3y = x – y = 5 and 5x + 2y = 23

It has at least one solution. It has different graphs.

Inconsistent Linear System > A linear system that has no solution. The slopes of an inconsistent linear system is equal.

Dependent Linear System  A linear system that has infinitely many solution.  It has identical graph The slopes and y-intercept is equal

Number of solutions Slopes and y - intercept One solutionDifferent slopes No solutionSame Slope Different y-intercept Infinitely many solutions Same Slope Same y-intercept

Bell Work Determine whether the statement is true or false. 1. A solution of a linear system is an ordered pair (x,y) 2. Graphically, the solution of an independent system is the point of intersection. 3. An independent system of equation has no solution.

4. The graph of an inconsistent system is identical. 5. A system of linear equation can either have one solution or no solution.

II. Answer the following. 6. Is (2,3) a solution of the system 3x + 4y = 18 2x – y = 1 ? 7. Is ( 1, -2 ) a solution of the system 3x – y = 14 2x + 5y = 8 ?

8. Is (-1,3) a solution of the system 4x – y = -5 2x + 5y = 13 ? 9. Is (0,0) a solution of the system 4x + 3y = 0 2x – y = 1 ? 10. Is (2,-3) a solution of the system y = 2x – 7 3x – y = 9 ?

Seatwork: Without graphing, determine whether the linear system is independent, inconsistent, or dependent. 1. y = -9x x + y = 6 y = 4x – 8 3x + y = x + y = 34. x – y = 3 2x + 2y = 6 x + y = 5

Seatwork: Without graphing, determine whether the linear system is independent, inconsistent, or dependent. 1. y = -9x x + y = 6 y = 4x – 8 3x + y = x + y = 34. x – y = 3 2x + 2y = 6 x + y = 5

5. 3x – y = 36. 2x – y = 4 2x + y = 2 x + y = 5 7. x + 2y = 68. 5x – 2y = 10 x – 2y = 3 3x + 2y = x – 5y = x + 2y = 6 8x – 10y = 12 y + 4x = -1

Seatwork

Homework Solve the following linear systems and indentify the type of system. 1. y = 7x + 13 and -21x + 3y = x – 2y = 7 and –x + 2y = 7

System of Linear Inequalities Consist of two or more linear inequalities in the same variable. Solution of a System of Linear Inequalities Is an ordered pair that is a solution for both linear system. Graph of a System of Linear Inequalities Is the graph of all solutions of a system.

y > -x – 2 and y ≤ 3x + 6

x > 1 and x <4

y > -3 and y < 4

Steps in Graphing a system of Linear Inequalities Step 1: Graph each inequality. Step 2: Find the intersection of the shaded part. The graph of a system is its intersection.

y ≥ -1, x > -2 and x + 2y ≤ 4

Seatwork

Homework Solve the following linear inequalities. 1. y ≤ x – 3 and y > -2x x -2

y ≤ x – 3 and y > -2x - 1

X -2

Problem Solving 1. There are 20 animals in a pen compose of dog and duck. If the number of their legs is 52, how many dogs are there? How many chickens are there?

2. A gardening company placed orders with nursery. One was fo