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Solving Sim. Equations Graphically

Solving Simple Sim. Equations by Substitution

Simultaneous Simultaneous EquationsEquations

Solving Simple Sim. Equations by elimination

Solving harder type Sim. equations

Graphs as Mathematical Models

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Starter QuestionsStarter Questions

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

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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

Simultaneous EquationsStraight Lines

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2 1 0 1 2 3 4

4

3

2

1

1

2

3

4Q. Find the equation of each line.

2Red : y x

2 1Blue : y x

(1,3)

Q. Write down the coordinates were they meet.

Simultaneous EquationsStraight Lines

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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.

Simultaneous EquationsStraight Lines

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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.

Simultaneous EquationsStraight Lines

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Now try Exercise 2Ch7 (page 84 )

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Starter QuestionsStarter Questions11. 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

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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 Lines

Simultaneous EquationsStraight Lines

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Per day 0 1 2 3 4 5Total Cost £ 0 60 120

Swinton Direct Car Hire

Per Day 0 1 2 3 4 5Total 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

Simultaneous EquationsStraight Lines

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Tota

l Cos

t £

Arnold

Swinton

Summarise data !

Who should I hire the car from?

Up to 2 days Swinton

Over 2 days Arnold

Simultaneous EquationsStraight Lines

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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.

Simultaneous EquationsStraight Lines

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Now try Exercise 3Ch7 (page 85 )

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Starter QuestionsStarter Questions

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

- 9yulate a) - 5x b) - 19y

+ 5x

S5 Int2

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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

Simultaneous EquationsStraight Lines

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Example 1

Solve the equations

y = 2x y = x+1

by substitution3 2 1 0 1 2 3

3

2

1

1

2

3

Simultaneous EquationsStraight Lines

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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)

Simultaneous EquationsStraight Lines

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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)

Simultaneous EquationsStraight Lines

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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 valuey = x +1 = 1.5 + 1 = 2.5y = -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

Simultaneous EquationsStraight Lines

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Now try Ex 4Ch7 (page88 )

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Starter QuestionsStarter Questions

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

+ 9wate a) 5y b) 12w

- 2y

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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 Lines

Simultaneous EquationsStraight Lines

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Example 1

Solve the equations

x + 2y = 14 x + y = 9

by elimination

Simultaneous EquationsStraight Lines

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Step 1: Label the equationsx + 2y = 14 (1)

x + y = 9 (2)

Step 2: Decide what you want to eliminateEliminate x by subtracting (2) from (1)x + 2y = 14 (1)x + y = 9 (2)

y = 5

Simultaneous EquationsStraight Lines

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Step 3: Sub into one of the equations to get other variable

Substitute y = 5 in (2)x + y = 9 (2)x + 5 = 9

The solution is x = 4 y = 5

Step 4: Check answers by substituting into both equations

x = 9 - 5x = 4

x + 2y = 14x + y = 9

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

Simultaneous EquationsStraight Lines

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Example 2

Solve the equations

2x - y = 11 x - y = 4

by elimination

Simultaneous EquationsStraight Lines

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Step 1: Label the equations2x - y = 11 (1)

x - y = 4 (2)

Step 2: Decide what you want to eliminateEliminate y by subtracting (2) from (1)2x - y = 11 (1) x - y = 4 (2)

x = 7

Simultaneous EquationsStraight Lines

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Step 3: Sub into one of the equations to get other variable

Substitute x = 7 in (2)x - y = 4 (2)7 - y = 4

The solution is x =7 y =3

Step 4: Check answers by substituting into both equations

y = 7 - 4y = 3

2x - y = 11 x - y = 4

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

Simultaneous EquationsStraight Lines

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Example 3

Solve the equations

2x - y = 6 x + y = 9

by elimination

Simultaneous EquationsStraight Lines

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Step 1: Label the equations2x - y = 6 (1)

x + y = 9 (2)

Step 2: Decide what you want to eliminateEliminate y by adding (1) and (2)2x - y = 6 (1) x + y = 9 (2)

3x = 15 x = 15 ÷ 3 = 5

Simultaneous EquationsStraight Lines

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Step 3: Sub into one of the equations to get other variable

Substitute x = 5 in (2)x + y = 9 (2)5 + y = 9

The solution is x = 5 y = 4Step 4:

Check answers by substituting into both equations

y = 9 - 5y = 4

2x - y = 6 x + y = 9

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

Simultaneous EquationsStraight Lines

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Now try Ex 5ACh7 (page89 )

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Starter QuestionsStarter Questions

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

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Learning IntentionLearning Intention Success CriteriaSuccess Criteria

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 Lines

Simultaneous EquationsStraight Lines

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Example 1

Solve the equations

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

by elimination

Simultaneous EquationsStraight Lines

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

Step 1: Label the equations2x + y = 9 (1)

x -3y = 1 (2)

Step 2: Decide what you want to eliminateEliminate y by :

7x = 28

6x + 3y = 27 (3) x - 3y = 1 (4)

x = 28 ÷ 7 = 4

Adding

(1) x3(2) x1

Simultaneous EquationsStraight Lines

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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

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)

Simultaneous EquationsStraight Lines

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Example 2

Solve the equations

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

by elimination

Simultaneous EquationsStraight Lines

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

Step 1: Label the equations3x + 2y = 13 (1)

2x + y = 8 (2)

Step 2: Decide what you want to eliminateEliminate y by :

-x = -3

3x + 2y = 13 (3) 4x + 2y = 16 (4) x = 3

Subtract

(1) x1(2) x2

Simultaneous EquationsStraight Lines

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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

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)

Simultaneous EquationsStraight Lines

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Now try Ex 5BCh7 (page90 )