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Simultaneous Equations. Solving Sim. Equations Graphically. Graphs as Mathematical Models. Solving Simple Sim. Equations by Substitution. Solving Simple Sim. Equations by elimination. Solving harder type Sim. equations. Starter Questions. Simultaneous Equations. S5 Int2. Straight Lines.

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## Simultaneous Equations

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**Simultaneous Equations**Solving Sim. Equations Graphically Graphs as Mathematical Models Solving Simple Sim. Equations by Substitution Solving Simple Sim. Equations by elimination Solving harder type Sim. equations**Simultaneous Equations**S5 Int2 Straight Lines Learning Intention Success Criteria • To solve simultaneous equations using graphical methods. • Interpret information from a line graph. • Plot line equations on a graph. • 3. Find the coordinates were 2 lines intersect ( meet)**Q. Find the equation**of each line. (1,3) Q. Write down the coordinates were they meet.**Q. Find the equation**of each line. Q. Write down the coordinates where they meet. (-0.5,-0.5)**Q. Plot the lines.**(1,1) Q. Write down the coordinates where they meet.**Now try Exercise 2**Ch7 (page 84 )**S5 Int2**Starter Questions 8cm 5cm**Simultaneous Equations**Straight Lines Learning Intention Success Criteria • To use graphical methods to solve real-life mathematical models • Draw line graphs given a table of points. • 2. Find the coordinates were 2 lines intersect ( meet)**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. 160 180 200 180 240 300**Summarise data !**Who should I hire the car from? Arnold Total Cost £ Up to 2 days Swinton Over 2 days Arnold Swinton Days**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.**Now try Exercise 3**Ch7 (page 85 )**S5 Int2**Starter Questions**Simultaneous Equations**S5 Int2 Straight Lines Learning Intention Success Criteria • To solve pairs of equations by substitution. 1. Apply the process of substitution to solve simple simultaneous equations.**Example 1**Solve the equations y = 2x y = x+1 by substitution**y = 2x**y = x+1 At the point of intersection y coordinates are equal: 2x = x+1 so we have 2x - x = 1 Rearranging we get : x = 1 Finally : Sub into one of the equations to get y value y = 2x = 2 x 1 = 2 y = x+1 = 1 + 1 = 2 OR The solution is x = 1 y = 2 or (1,2)**Example 1**Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5)**y = x +1**y =-x+ 4 The solution is x = 1.5 y = 2.5 (1.5,2.5) At the point of intersection y coordinates are equal: x+1 = -x+4 so we have 2x = 4 - 1 Rearranging we get : 2x = 3 x = 3 ÷ 2 = 1.5 Finally : Sub into one of the equations to get y value y = x +1 = 1.5 + 1 = 2.5 y = -x+4 = -1.5 + 4 = 2 .5 OR**Now try Ex 4**Ch7 (page88 )**Simultaneous Equations**Straight Lines Learning Intention Success Criteria • To solve simultaneous equations of 2 variables. • Understand the term simultaneous equation. • Understand the process for solving simultaneous equation of two variables. • 3. Solve simple equations**Example 1**Solve the equations x + 2y = 14 x + y = 9 by elimination**Step 1: Label the equations**x + 2y = 14 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate x by subtracting (2) from (1) x + 2y = 14 (1) x + y = 9 (2) y = 5**Step 3: Sub into one of the equations to get other variable**Substitute y = 5 in (2) x + y = 9 (2) x + 5 = 9 x = 9 - 5 The solution is x = 4 y = 5 x = 4 Step 4: Check answers by substituting into both equations ( 4 + 10 = 14) x + 2y = 14 x + y = 9 ( 4 + 5 = 9)**Example 2**Solve the equations 2x - y = 11 x - y = 4 by elimination**Step 1: Label the equations**2x - y = 11 (1) x - y = 4 (2) Step 2: Decide what you want to eliminate Eliminate y by subtracting (2) from (1) 2x - y = 11 (1) x - y = 4 (2) x = 7**Step 3: Sub into one of the equations to get other variable**Substitute x = 7 in (2) x - y = 4 (2) 7 - y = 4 y = 7 - 4 The solution is x =7 y =3 y = 3 Step 4: Check answers by substituting into both equations ( 14 - 3 = 11) 2x - y = 11 x - y = 4 ( 7 - 3 = 4)**Example 3**Solve the equations 2x - y = 6 x + y = 9 by elimination**Step 1: Label the equations**2x - y = 6 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate y by adding (1) and (2) 2x - y = 6 (1) x + y = 9 (2) x = 15 ÷ 3 = 5 3x = 15**Step 3: Sub into one of the equations to get other variable**Substitute x = 5 in (2) x + y = 9 (2) 5 + y = 9 y = 9 - 5 The solution is x = 5 y = 4 y = 4 Step 4: Check answers by substituting into both equations ( 10 - 4 = 6) 2x - y = 6 x + y = 9 ( 5 + 4 = 9)**Now try Ex 5A**Ch7 (page89 )**Simultaneous Equations**Straight Lines Learning Intention Success Criteria • To solve harder simultaneous equations of 2 variables. 1. Apply the process for solving simultaneous equations to harder examples.**Example 1**Solve the equations 2x + y = 9 x - 3y = 1 by elimination**Step 1: Label the equations**2x + y = 9 (1) x -3y = 1 (2) Step 2: Decide what you want to eliminate Adding Eliminate y by : (1) x3 2x + y = 9 x -3y = 1 6x + 3y = 27 (3) x - 3y = 1(4) (2) x1 7x = 28 x = 28 ÷ 7 = 4**Step 3: Sub into one of the equations to get other variable**Substitute x = 4 in equation (1) 2 x 4 + y = 9 y = 9 – 8 y = 1 The solution is x = 4 y = 1 Step 4: Check answers by substituting into both equations ( 8 + 1 = 9) 2x + y = 9 x -3y = 1 ( 4 - 3 = 1)**Example 2**Solve the equations 3x + 2y = 13 2x + y = 8 by elimination**Step 1: Label the equations**3x + 2y = 13 (1) 2x + y = 8 (2) Step 2: Decide what you want to eliminate Subtract Eliminate y by : (1) x1 3x + 2y = 13 2x + y = 8 3x + 2y = 13 (3) 4x + 2y = 16(4) (2) x2 -x = -3 x = 3**Step 3: Sub into one of the equations to get other variable**Substitute x = 3 in equation (2) 2 x 3 + y = 8 y = 8 – 6 y = 2 The solution is x = 3 y = 2 Step 4: Check answers by substituting into both equations ( 9 + 4 = 13) 3x + 2y = 13 2x + y = 8 ( 6 + 2 = 8)**Now try Ex 5B**Ch7 (page90 )

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