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

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

21 Apr 202321 Apr 2023

Starter QuestionsStarter Questionsw

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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

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.com 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

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Simultaneous EquationsS5

Int2 Straight Lines

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2 1 0 1 2 3 4

Q. Find the equation of each line.

2Red : y x

2 1Blue : y x

Q. Write down the coordinates were they meet.

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Int2 Straight Lines

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2 1.5 1 0.5 0 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.

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Int2 Straight Lines

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2 1.5 1 0.5 0 0.5 1 1.5 2

Q. Plot the lines.

2 1Blue : y x

Q. Write down the coordinates where they meet.

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Int2 Straight Lines

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Now try Exercise 2Ch7 (page 84 )

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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

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

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

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Int2 Straight Lines

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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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Int2 Straight Lines

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Arnold

Swinton

Summarise data !

Who should I hire the car from?

Up to 2 days Swinton

Over 2 days Arnold

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Int2 Straight 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.

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Int2 Straight Lines

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Now try Exercise 3Ch7 (page 85 )

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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

ulate a) - 5x b) - 19y

S5 Int2

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1. To solve pairs of equations by substitution.

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

Straight LinesS5

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Int2 Straight Lines

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Example 1

Solve the equations

y = 2x y = x+1

by substitution

3 2 1 0 1 2 3

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Int2 Straight 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)

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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

(1.5, 2.5)

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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

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Int2 Straight Lines

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Now try Ex 4Ch7 (page88 )

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21 Apr 202321 Apr 2023

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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

ate a) 5y b) 12w

S5 Int2

21 Apr 202321 Apr 2023ww

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1. To solve simultaneous equations of 2 variables.

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

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Int2 Straight Lines

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Example 1

Solve the equations

x + 2y = 14 x + y = 9

by elimination

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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)

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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 + 2y = 14x + y = 9

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

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Int2 Straight Lines

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Example 2

Solve the equations

2x - y = 11 x - y = 4

by elimination

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Int2 Straight Lines

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

x - y = 4 (B)

Eliminate y by subtracting (B) from (A)2x - y = 11 (A) x - y = 4 (B)

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Int2 Straight Lines

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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

2x - y = 11 x - y = 4

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

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Int2 Straight Lines

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Example 3

Solve the equations

2x - y = 6 x + y = 9

by elimination

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Int2 Straight Lines

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

x + y = 9 (B)

Eliminate y by adding (A) from (B)

2x - y = 6 (A) x + y = 9 (B)

3x = 15 x = 15 ÷ 3 = 5

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Int2 Straight Lines

21 Apr 202321 Apr 2023

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

2x - y = 6 x + y = 9

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

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Int2 Straight Lines

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Now try Ex 5ACh7 (page89 )

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21 Apr 202321 Apr 2023

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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

21 Apr 202321 Apr 2023ww

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1. To solve harder simultaneous equations of 2 variables.

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

Straight LinesS5

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Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 1

Solve the equations

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

by elimination

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Int2 Straight Lines

21 Apr 202321 Apr 2023

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

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

x -3y = 1 (B)

Eliminate y by :

7x = 28

6x + 3y = 27 (C) x - 3y = 1 (D)

x = 28 ÷ 7 = 4

Adding

(A) x3(B) x1

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Int2 Straight Lines

21 Apr 202321 Apr 2023

Substitute x = 4 in equation (A)

2 x 4 + y = 9

y = 9 – 8

The solution is x = 4 y = 1

Step 4:

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

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

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Int2 Straight Lines

21 Apr 202321 Apr 2023

Example 2

Solve the equations

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

by elimination

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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)

Eliminate y by :

-x = -3

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

Subtract

(A) x1(B) x2

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Int2 Straight Lines

21 Apr 202321 Apr 2023

Substitute x = 3 in equation (B)

2 x 3 + y = 8

y = 8 – 6

The solution is x = 3 y = 2Step 4:

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

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

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Int2 Straight Lines

21 Apr 202321 Apr 2023

Now try Ex 5BCh7 (page90 )

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

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