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19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
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
19 Apr 202319 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.
Simultaneous EquationsStraight Lines
19 Apr 202319 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.
Simultaneous EquationsStraight Lines
19 Apr 202319 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.
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Now try Exercise 2Ch7 (page 84 )
19 Apr 202319 Apr 2023
Starter QuestionsStarter Questions
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
19 Apr 202319 Apr 2023
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
19 Apr 202319 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
Simultaneous EquationsStraight Lines
19 Apr 202319 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
Simultaneous EquationsStraight Lines
19 Apr 202319 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.
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Now try Exercise 3Ch7 (page 85 )
19 Apr 202319 Apr 2023
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
- 9y
ulate a) - 5x b) - 19y
+ 5x
S5 Int2
19 Apr 202319 Apr 2023
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
19 Apr 202319 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
Simultaneous EquationsStraight Lines
19 Apr 202319 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)
Simultaneous EquationsStraight Lines
19 Apr 202319 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)
Simultaneous EquationsStraight Lines
19 Apr 202319 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
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Now try Ex 4Ch7 (page88 )
19 Apr 202319 Apr 2023
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
+ 9w
ate a) 5y b) 12w
- 2y
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
Example 1
Solve the equations
x + 2y = 14 x + y = 9
by elimination
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Step 1: Label the equationsx + 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
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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 - 5
x = 4
x + 2y = 14x + y = 9
( 4 + 10 = 14)( 4 + 5 = 9)
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Example 2
Solve the equations
2x - y = 11 x - y = 4
by elimination
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Step 1: Label the equations2x - 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
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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 - 4
y = 3
2x - y = 11 x - y = 4
( 14 - 3 = 11)( 7 - 3 = 4)
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Example 3
Solve the equations
2x - y = 6 x + y = 9
by elimination
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Step 1: Label the equations2x - 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)
3x = 15 x = 15 ÷ 3 = 5
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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 - 5
y = 4
2x - y = 6 x + y = 9
( 10 - 4 = 6)( 5 + 4 = 9)
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
Now try Ex 5ACh7 (page89 )
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
Example 1
Solve the equations
2x + y = 9 x - 3y = 1
by elimination
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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 eliminate
Eliminate y by :
7x = 28
6x + 3y = 27 (3) x - 3y = 1 (4)
x = 28 ÷ 7 = 4
Adding
(1) x3(2) x1
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
Example 2
Solve the equations
3x + 2y = 132x + y = 8
by elimination
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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 eliminate
Eliminate y by :
-x = -3
3x + 2y = 13 (3) 4x + 2y = 16 (4) x = 3
Subtract
(1) x1(2) x2
Simultaneous EquationsStraight Lines
19 Apr 202319 Apr 2023
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
19 Apr 202319 Apr 2023
Now try Ex 5BCh7 (page90 )
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