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Solve Linear Systems by Graphing Solve Linear Systems by Substitution Solve Linear Systems by Elimination Adding or Subt

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## Solve Linear Systems by Graphing Solve Linear Systems by Substitution Solve Linear Systems by Elimination Adding or Subt

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**Systems of**Linear Equation Solve Linear Systems by Graphing Solve Linear Systems by Substitution Solve Linear Systems by Elimination Adding or Subtracting Multiplying First Solve Special Type of Linear Systems Solve systems of Linear Inequalities**Solve Linear**Systems by Graphing**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. Can you name a pair that satisfies both equation? 4 3 2 1 0 -1 2. Complete the table below for 2x – y = 5. -5 -3 -1 1 3 5**x + y = 4**2x – y = 5 System of Linear Equations • Linear system • Consist of two or more linear equations in the same variables • x + 2y = 7 and 3x – 2y = 5 Solution of a System of Linear Equations (3, 1) Is an ordered pair that satisfies each of the equation in the systems**4**3 2 1 0 -1 Point of intersection (3,1) -5 -3 -1 1 3 5**1) -5x + y = 0 and 5x + y = 10**y = -5x +10 1.Graph y = 5x 3.Check 2.Identify the point of intersection y = -5x + 10 5 = -5(1) + 10 5 = -5 +10 5 = 5 y = 5x 5 = 5(1) 5 = 5 (1, 5) Point of intersection (1,5) is a solution of the Linear system**2) x – y = 5 and 3x + y = 3**y = -3x + 3 1.Graph y = x - 5 2.Identify the point of intersection 3.Check y = -3x + 3 -3 = -3(2) + 3 -3 = -6 + 3 -3 = -3 y = x – 5 -3 = 2 – 5 -3 = -3 (2, -3) Point of intersection (2, -3) is a solution of the Linear system**It has at least one solution.**Consistent Independent system of Linear equation It has different graphs.**F.Y.I.:**Inconsistent system of linear equations – does not have a solution. Dependent system of linear equations – equations with identical graph.**Steps in Solving for Systems by Graphing**1. Graph both equations in the same coordinate plane. 2. Identify the point of intersection. 3. Check the coordinates algebraically by substituting it in to each equation.**3) x + y = 4 and 2x – y = 5**y = 2x - 5 1.Graph y = -x + 4 2.Identify the point of intersection 3.Check y = 2x - 5 1 = 2(3) – 5 1 = 6 - 5 1 = 1 y = -x + 4 1 = -(3) + 4 1 = 1 (3, 1) Point of intersection (3,1) is a solution of the Linear system**4) x – y = 1 and x + y = 3**y = -x + 3 1.Graph y = x - 1 3.Check 2.Identify the point of intersection y = -x + 3 1 = -(2) + 3 1 = -2 + 3 1 = 1 y = x – 1 1 = 2 – 1 1 = 1 (2, 1) Point of intersection (2, 1) is a solution of the Linear system**Homework**Solve the following linear systems by graphing. • y = -x + 3 and y = x + 1 • 3x + y = 15 and y = -15**2.3x + y = 15**y = -15 • y = -x + 3 y = x + 1**Solve Linear**Systems by Substitution**A gardening company**placed orders with a nursery. One was for 13 bushes and 4trees, and totaledP487. 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: • 3(2x + 3) • -4(3x – 4) Can you solve the system of linear equation by substitution? Substitute x – 3 for y and simplify the following: • 5y • 3y + 2 • 2(y+3) • Solve for x and y • 1. y = 2x + 1 and 3x + 2y = 9**Eq. 1**Eq. 2 y=2x + 1 and 3x + 2y = 9 Eq. 1 is already solved for y. Solve for a variable 3x + 2y = 9 eq. 2 2. Substitute Eq.1 to Eq. 2 3x + 2(2x + 1) = 9 3x + 4x + 2 = 9 7x = 9-2 3. Substitute value of x to eq. 1 7x = 7 x = 1 y = 2x + 1 eq. 1 y = 2(1) + 1 y = 2 + 1 The solution is (1, 3). y = 3**Eq. 1**Eq. 2 a + b = 7 and 3a + 2b = 16 1. Solve for a variable a + b = 7 eq. 1 b = -a + 7 3a + 2b = 16 eq. 2 3a + 2(-a + 7) = 16 2. Substitute Eq.1 to Eq. 2 3a – 2a + 14 = 16 a = 16 - 14 a = 2 3. Substitute value of a to eq. 1 b = -a + 7 eq. 1 b = -(2) + 7 The solution is (2, 5). b = -2 + 7 b = 5**Steps in Solving for Systems by Substitution**1. Solve for one variable using one of the equations. 2. Substitute the expression from step 1 into other equation and solve. 3. Substitute the value from step 2 into the expression of step 1 and solve.**Eq. 1**Eq. 2 x – 2y = -6 and 4x + 6y = 4 Solve for a variable x – 2y = -6 x = 2y - 6 4x + 6y = 4 eq. 2 2. Substitute Eq.1 to Eq. 2 4(2y – 6) + 6y = 4 8y – 24 + 6y = 4 14y = 4 + 24 3. Substitute value of x to eq. 1 14y = 28 y = 2 x = 2y - 6 eq. 1 x = 2(2) - 6 x = 4 - 6 The solution is (-2, 2). x = -2**Eq. 1**Eq. 2 m = 2n + 5 and 3n + m = 10 Eq. 1 is already solved for m. 1. Solve for a variable 3n + m = 10 eq. 2 3n + (2n + 5) = 10 2. Substitute Eq.1 to Eq. 2 3n + 2n + 5 = 10 5n = 10 - 5 5n = 5 n = 1 3. Substitute value of x to eq. 1 m = 2n + 5 eq. 1 The solution is (1, 7). m = 2(1) + 5 m = 2 + 5 m = 7**Homework**Solve the following linear systems by substitution • x = y + 3 and 2x – y = 5 • 11a – 7b = -14 and a- 2b =-4**Seatwork**Solve each of the following linear systems by substitution and graph. • 2x – y = 4 and 3x + 2y = -4 • 3x = 2y + 5 and y + 1 = 0 • x + 2y = 3 and x – y = 6 • 3x = 2y + 8 and y = 3x**Solve Linear**Systems by Elimination**Getting ready!**Solve for the following: (-2x - 4y) + (-4x + 4y) (3x -2y) + (3x + 3y) = (x + y) - (x - 4y) (3x - 4y) + (-3x + 2y) =**Developing Skills**The solution is (1, 2) Add the following equation. x + 2y = 5 3x – 2y = -1 y 4x = 4 x + 2y = 5 1 + 2y = 5 2y = 5-1 2y = 4 y = 2 x = 1 Which variable is eliminated? Solve for the remaining variable.**The solution**is (4, -2) 2a – 3b = 14 a + 3b = -2 Add the following equation. 3a = 12 b a = 4 a + 3b = -2 4 + 3b = -2 3b = -2 - 4 3b = -6 b = -2 Solve for the remaining variable. Which variable is eliminated?**Developing Skills**The solution is (1, 3) Add the following equation. 2x + 3y = 11 2x - 5y = 13 x 8y = 24 2x + 3y = 11 2x + 3(3) = 11 2x = 11 - 9 2x = 2 x = 1 y = 3 Which variable is eliminated? Solve for the remaining variable.**Steps in Solving for Systems by adding and subtracting**1. Add or subtract the equations to eliminate one variable. 2. Solve for one variable. 3. Substitute the value from step 2 into one original equation and solve.**The solution**is (12,-4) 2a – 3b = 26 -2a - 3b = -2 Add the following equation. -6b = 24 b b= -4 2a - 3b = 26 2a – 3(-4) = 26 2a = 26 - 12 2a = 24 a = 12 Solve for the remaining variable. Which variable is eliminated?**Solve Linear System by**MULTIPLYING FIRST**5x + 2y = 16**3x – 4y = 20 (5x + 2y = 16)2 10x + 4y = 32 3x – 4y = 20 Can we eliminate a variable by adding and subtracting?**10x + 4y = 32**The solution is (4, -2) 3x – 4y = 20 13x = 52 x = 4 3x – 4y = 20 3(4) – 4y = 20 – 4y = 20 -12 – 4y = 8 y = -2**6x + 5y = 19**6x + 5y = 19 ( )-3 2x + 3y = 5 -6x - 9y = -15 -4y = 4 6x + 5y = 19 y = -1 6x +5(-1) = 19 6x = 19 + 5 6x = 24 The solution is (4, -1) x = 4**( ) 2**8x + 10y = 70 4x + 5y = 35 ( ) 5 -3x + 2y = -9 -15x+10y = -45 23x = 115 The solution is (4, -1) x = 5 4x + 5y = 35 4(5) +5y = 35 5y = 35 - 20 5y = 15 y = 3**Seatwork**Solve each of the following linear systems by Elimination. ) 6x – 2y = 1 -2x + 3y = -5 2.) 2x + 5y = 3 3x + 10y = -3**Homework**Solve the following linear systems by Elimination • 3x - 7y = 5 and 9y= 5x + 5 • 3a + 2b = 4 and 2b =8 – 5a**Short Quiz**Solve the following linear systems by Elimination • x + 4y = 22 and 4x – y = 3 • 2x – 3y = 10 and x + 3y = -8 • 3x – y = 5 and 5x + 2y = 23**Solve Special Type of Linear**Systems**It has at least one solution.**Consistent Independent system of Linear equation It has different graphs.**Inconsistent Linear System**The slopes of an inconsistent linear system is equal. > A linear system that has no solution. 3x + 2y = 10 3x + 2y = 2 No solution**Dependent Linear System**x – 2y = -4 y = (1/2)x + 2 • A linear system that has infinitely many solution. • It has identical graph The slopes and y-intercept is equal Infinitely many solution**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 ?