26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations Solving Simple Sim. Equations by elimination Solving harder type Sim. equations S5 Int2 Graphs as Mathematical Models

26-Dec-15 Starter Questions Starter Questions S5 Int2

26-Dec-15 Learning Intention Success Criteria 1.To solve simultaneous equations using graphical methods. Simultaneous Equations 1.Interpret information from a line graph. 2.Plot line equations on a graph. 3.Find the coordinates were 2 lines intersect ( meet) Straight Lines S5 Int2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Find the equation of each line. (1,3) Q. Write down the coordinates were they meet.

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Find the equation of each line. (-0.5,-0.5) Q. Write down the coordinates where they meet.

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet.

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Exercise 2 Ch7 (page 84 )

26-Dec-15 Starter Questions Starter Questions S5 Int2 5cm 8cm

26-Dec-15 Learning Intention Success Criteria 1.To use graphical methods to solve real-life mathematical models Simultaneous Equations 1.Draw line graphs given a table of points. 2.Find the coordinates were 2 lines intersect ( meet) Straight Lines S5 Int2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 We can use straight line theory to work out real-life problems especially useful when trying to work out hire charges. Q.I need to hire a car for a number of days. Below are the hire charges charges for two companies. Complete tables and plot values on the same graph

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Days Total Cost £ A r n o l d S w i n t o n Summarise data ! Who should I hire the car from? Up to 2 days Swinton Over 2 days Arnold

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Key steps 1. Fill in tables 2. Plot points on the same graph ( pick scale carefully) 3. Identify intersection point ( where 2 lines meet) 4. Interpret graph information.

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Exercise 3 Ch7 (page 85 )

26-Dec-15 Starter Questions Starter Questions S5 Int2

26-Dec-15 Learning Intention Success Criteria 1.To solve pairs of equations by substitution. Simultaneous Equations 1.Apply the process of substitution to solve simple simultaneous equations. Straight Lines S5 Int2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations y = 2x y = x+1 by substitution

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 At the point of intersection y coordinates are equal: 2x = x+1 Rearranging we get : 2x - x = 1 x = 1 Finally : Sub into one of the equations to get y value y = 2x = 2 x 1 = 2 OR y = x+1 = = 2 so we have y = 2x y = x+1 The solution is x = 1 y = 2 or (1,2)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 At the point of intersection y coordinates are equal: x+1 = -x+4 Rearranging we get : 2x = x = 3 Finally : Sub into one of the equations to get y value y = x +1 = = 2.5 y = -x+4 = = 2.5 so we have y = x +1 y =-x+ 4 The solution is x = 1.5 y = 2.5 (1.5,2.5) x = 3 ÷ 2 = 1.5 OR

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 4 Ch7 (page88 )

26-Dec-15 Starter Questions Starter Questions S5 Int2

26-Dec-15 Learning Intention Success Criteria 1.To solve simultaneous equations of 2 variables. Simultaneous Equations 1.Understand the term simultaneous equation. 2.Understand the process for solving simultaneous equation of two variables. 3.Solve simple equations Straight Lines S5 Int2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations x + 2y = 14 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate x by subtracting (B) from (A) x + 2y = 14 (A) x + y = 9 (B) y = 5

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute y = 5 in (B) x + y = 9 (B) x + 5 = 9 The solution is x = 4 y = 5 Step 4:Check answers by substituting into both equations x = x = 4 x + 2y = 14 x + y = 9 ( = 14) ( = 9)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations 2x - y = 11 (A) x - y = 4 (B) Step 2: Decide what you want to eliminate Eliminate y by subtracting (B) from (A) 2x - y = 11 (A) x - y = 4 (B) x = 7

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (B) x - y = 4 (B) 7 - y = 4 The solution is x =7 y =3 Step 4:Check answers by substituting into both equations y = y = 3 2x - y = 11 x - y = 4 ( = 11) ( = 4)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations 2x - y = 6 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate y by adding (A) from (B) 2x - y = 6 (A) x + y = 9 (B) 3x = 15 x = 15 ÷ 3 = 5

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 5 in (B) x + y = 9 (B) 5 + y = 9 The solution is x = 5 y = 4 Step 4: Check answers by substituting into both equations y = y = 4 2x - y = 6 x + y = 9 ( = 6) ( = 9)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 5A Ch7 (page89 )

26-Dec-15 Starter Questions Starter Questions S5 Int2

26-Dec-15 Learning Intention Success Criteria 1.To solve harder simultaneous equations of 2 variables. Simultaneous Equations 1.Apply the process for solving simultaneous equations to harder examples. Straight Lines S5 Int2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 2x + y = 9 x -3y = 1 Step 1: Label the equations 2x + y = 9(A) x -3y = 1(B) Step 2: Decide what you want to eliminate Eliminate y by : 7x = 28 6x + 3y = 27 (C) x - 3y = 1 (D) x = 28 ÷ 7 = 4 Adding (A) x3 (B) x1

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 4 in equation (A) 2 x 4 + y = 9 y = 9 – 8 The solution is x = 4 y = 1 Step 4: Check answers by substituting into both equations y = 1 2x + y = 9 x -3y = 1 ( = 9) ( = 1)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 2 Solve the equations 3x + 2y = 13 2x + y = 8 by elimination

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 3x + 2y = 13 2x + y = 8 Step 1: Label the equations 3x + 2y = 13(A) 2x + y = 8(B) Step 2: Decide what you want to eliminate Eliminate y by : -x = -3 3x + 2y = 13 (C) 4x + 2y = 16 (D) x = 3 Subtract (A) x1 (B) x2

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 3 in equation (B) 2 x 3 + y = 8 y = 8 – 6 The solution is x = 3 y = 2 Step 4: Check answers by substituting into both equations y = 2 3x + 2y = 13 2x + y = 8 ( = 13) ( = 8)

Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 5B Ch7 (page90 )