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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 )