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Page 1: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

21 Apr 202321 Apr 2023

Solving Sim. Equations Graphically

Solving Simple Sim. Equations by Substitution

Simultaneous Simultaneous EquationsEquations

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Solving Simple Sim. Equations by elimination

Solving harder type Sim. equations

S5 Int2

Graphs as Mathematical Models

Page 2: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Starter QuestionsStarter Questionsw

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2. T

11. A car reduced in value by 33 % in one year.

3 I f its initial price was £12 000. How much is it now.

idy up the expression

- 6e + 5d - 7d +6e

3. Calculate a) 13y b) 12x

+ 4x- 5y

S5 Int2

Page 3: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

21 Apr 202321 Apr 2023ww

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1. To solve simultaneous equations using graphical methods.

Simultaneous Equations

1.1. Interpret information from a Interpret information from a line graph.line graph.

2.2. Plot line equations on a Plot line equations on a graph.graph.

3.3. Find the coordinates were 2 Find the coordinates were 2 lines intersect ( meet)lines intersect ( meet)

Straight LinesS5

Int2

Page 4: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

2 1 0 1 2 3 4

4

3

2

1

1

2

3

4

Q. Find the equation of each line.

2Red : y x

2 1Blue : y x

(1,3)

Q. Write down the coordinates were they meet.

Page 5: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

2 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0.5

1

1.5

2

Q. Find the equation of each line.

Red : y x

1Blue : y x

(-0.5,-0.5)Q. Write down the coordinates where they meet.

Page 6: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

2 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0.5

1

1.5

2

Q. Plot the lines.

y x

2 1Blue : y x

(1,1)

Q. Write down the coordinates where they meet.

Page 7: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Exercise 2Ch7 (page 84 )

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Page 8: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Starter QuestionsStarter Questionsw

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11. A house has increased in value by 12 % in one year.

2 I f its initial price was £96 000. How much is it now.

2. Find the area of the shape

3. Calculate 55% of 360

S5 Int2

5cm

8cm

Page 9: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

21 Apr 202321 Apr 2023ww

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.com Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1. To use graphical methods to solve real-life mathematical models

Simultaneous Equations

1.1. Draw line graphs given a Draw line graphs given a table of points.table of points.

2.2. Find the coordinates were 2 Find the coordinates were 2 lines intersect ( meet)lines intersect ( meet)

Straight LinesS5

Int2

Page 10: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Per day 0 1 2 3 4 5

Total Cost £ 0 60 120

Swinton Direct Car Hire

Per Day 0 1 2 3 4 5

Total Cost £ £100 120 140

Anrnold Palmer Car Hire

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

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023Days

Tota

l C

ost

£

Arnold

Swinton

Summarise data !

Who should I hire the car from?

Up to 2 days Swinton

Over 2 days Arnold

Page 12: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Key steps1. 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.

Page 13: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Exercise 3Ch7 (page 85 )

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Page 14: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Starter QuestionsStarter Questionsw

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2. Rearrange the equations below into y =

1. Write down the gradient and were line crosses y- axis.

y = - 2x +6

(a) y + 2x = 8

(b) 4x + 2y =10

3. Calc

- 9y

ulate a) - 5x b) - 19y

+ 5x

S5 Int2

Page 15: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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1. To solve pairs of equations by substitution.

Simultaneous Equations

1.1. Apply the process of Apply the process of substitution to solve simple substitution to solve simple simultaneous equations.simultaneous equations.

Straight LinesS5

Int2

Page 16: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 1

Solve the equations

y = 2x y = x+1

by substitution

3 2 1 0 1 2 3

3

2

1

1

2

3

Page 17: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

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 = 1 + 1 = 2

so we have

y = 2x y = x+1

The solution is x = 1 y = 2 or (1,2)

Page 18: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 1

Solve the equations

y = x + 1 x + y = 4

by substitution

0 0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

2.5

3

3.5

4

(1.5, 2.5)

Page 19: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

At the point of intersection y coordinates are equal:

x+1 = -x+4 Rearranging we get : 2x = 4 - 1

2x = 3

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

so we have

y = x +1y =-x+ 4

The solution is x = 1.5 y = 2.5 (1.5,2.5)

x = 3 ÷ 2 = 1.5

OR

Page 20: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Ex 4Ch7 (page88 )

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Page 21: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Starter QuestionsStarter Questionsw

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2. Rearrange the equations below into y =

1. A house appreciates by 20% in one year.

I f its initial price was £60 000. How much is it now.

(a) y + x = 2

(b) x + y =5

3. Calcul

+ 9w

ate a) 5y b) 12w

- 2y

S5 Int2

Page 22: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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1. To solve simultaneous equations of 2 variables.

Simultaneous Equations

1.1. Understand the term Understand the term simultaneous equation.simultaneous equation.

2.2. Understand the process Understand the process for solving simultaneous for solving simultaneous equation of two variables.equation of two variables.

3.3. Solve simple equationsSolve simple equations

Straight LinesS5

Int2

Page 23: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 1

Solve the equations

x + 2y = 14 x + y = 9

by elimination

Page 24: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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Step 1: Label the equationsx + 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

Page 25: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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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 = 9 - 5

x = 4

x + 2y = 14x + y = 9

( 4 + 10 = 14)( 4 + 5 = 9)

Page 26: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 2

Solve the equations

2x - y = 11 x - y = 4

by elimination

Page 27: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Step 1: Label the equations2x - 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

Page 28: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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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 = 7 - 4

y = 3

2x - y = 11 x - y = 4

( 14 - 3 = 11)( 7 - 3 = 4)

Page 29: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 3

Solve the equations

2x - y = 6 x + y = 9

by elimination

Page 30: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Step 1: Label the equations2x - 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

Page 31: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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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 = 4Step 4:

Check answers by substituting into both equations

y = 9 - 5

y = 4

2x - y = 6 x + y = 9

( 10 - 4 = 6)( 5 + 4 = 9)

Page 32: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Ex 5ACh7 (page89 )

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Starter QuestionsStarter Questionsw

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2. Tidy up the expressions

1. A fridge is increased by 4%.

I f its old price was £240. Find the new price.

(a) 2y - x - 3y +2x

(b) 4x - 5y + 6x +8y

3. Solve the equations : x + y = 6

x - y = 4

S5 Int2

Page 34: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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1. To solve harder simultaneous equations of 2 variables.

Simultaneous Equations

1.1. Apply the process for solving Apply the process for solving simultaneous equations to simultaneous equations to harder examples.harder examples.

Straight LinesS5

Int2

Page 35: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 1

Solve the equations

2x + y = 9 x - 3y = 1

by elimination

Page 36: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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2x + y = 9 x -3y = 1

Step 1: Label the equations2x + 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

Page 37: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

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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

( 8 + 1 = 9)( 4 - 3 = 1)

Page 38: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 2

Solve the equations

3x + 2y = 132x + y = 8

by elimination

Page 39: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

3x + 2y = 13 2x + y = 8

Step 1: Label the equations3x + 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

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Simultaneous EquationsS5

Int2 Straight Lines

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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 = 2Step 4:

Check answers by substituting into both equations

y = 2

3x + 2y = 132x + y = 8

( 9 + 4 = 13)( 6 + 2 = 8)

Page 41: 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations  Solving Simple

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Simultaneous EquationsS5

Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Ex 5BCh7 (page90 )

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