1. Simultaneous Equations
S4 General
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
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Starter Questions
S4 General
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Simultaneous Equations
S4 General
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)
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Q. Find the equation
of each line.
(1,3)
Q. Write down the
coordinates were
they meet.
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Q. Find the equation
of each line.
Q. Write down the
coordinates
where they meet.
(-0.5,-0.5)
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Q. Plot the lines.
(1,1)
Q. Write down the
coordinates
where they meet.
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Now try Exercise 2
Ch7 (page 84 )
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Starter Questions
S4 General
8cm
5cm
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Simultaneous Equations
S4 General
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)
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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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11. Who should I hire the car from?
Summarise data !
Who should I hire the car from?
Total Cost £
Arnold
Swinton
Up to 2 days Swinton
Over 2 days Arnold
Days
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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.
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Now try Exercise 3
Ch7 (page 85 )
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Starter Questions
S4 General
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Simultaneous Equations
S4 General
Straight Lines
Learning Intention
Success Criteria
To solve pairs of equations by substitution.
1. Apply the process of substitution to solve simple simultaneous equations.
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Example 1
Solve the equations
y = 2x
y = x+1
by substitution
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At the point of intersection y coordinates are equal:
y = 2x
y = x+1
so we have
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 = = 2
The solution is x = 1 y = 2 or (1,2)
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Example 1
Solve the equations
y = x + 1
x + y = 4
by substitution
(1.5, 2.5)
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The solution is x = 1.5 y = 2.5 (1.5,2.5)
At the point of intersection y coordinates are equal:
so we have
x+1 = -x+4
y = x +1
y =-x+ 4
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 = = 2.5 OR
y = -x+4 = = 2 .5
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Now try Ex 4
Ch7 (page88 )
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Starter Questions
S4 General
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Simultaneous Equations
S4 General
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
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Example 1
Solve the equations
x + 2y = 14
x + y = 9
by elimination
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Step 1: Label the equations
x + 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
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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
x = 9 - 5
The solution is x = 4
y = 5
x = 4
Step 4: Check answers by substituting into both equations
x + 2y = 14
x + y = 9
( = 14)
( = 9)
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Example 2
Solve the equations
2x - y = 11
x - y = 4
by elimination
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Step 1: Label the equations
2x - 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
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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
y = 7 - 4
The solution is x =7
y =3
y = 3
Step 4: Check answers by substituting into both equations
2x - y = 11
x - y = 4
( = 11)
( = 4)
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Example 3
Solve the equations
2x - y = 6
x + y = 9
by elimination
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Step 1: Label the equations
2x - 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
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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
y = 9 - 5
The solution is x = 5
y = 4
y = 4
Step 4:
Check answers by substituting into both equations
2x - y = 6
x + y = 9
( = 6)
( = 9)
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Now try Ex 5A
Ch7 (page89 )
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Starter Questions
S4 General
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Simultaneous Equations
S4 General
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.
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Example 1
Solve the equations
2x + y = 9
x - 3y = 1
by elimination
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Step 1: Label the equations
2x + y = 9 (A)
x -3y = 1 (B)
Step 2: Decide what you want to eliminate
Adding
Eliminate y by :
2x + y = 9
x -3y = 1
(A) x3
6x + 3y = 27 (C)
x - 3y = 1 (D)
(B) x1
7x = 28
x = 28 ÷ 7 = 4
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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
y = 1
The solution is x = 4
y = 1
Step 4:
Check answers by substituting into both equations
2x + y = 9
x -3y = 1
( = 9)
( = 1)
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Example 2
Solve the equations
3x + 2y = 13
2x + y = 8
by elimination
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Step 1: Label the equations
3x + 2y = 13 (A)
2x + y = 8 (B)
Step 2: Decide what you want to eliminate
Subtract
Eliminate y by :
3x + 2y = 13
2x + y = 8
(A) x1
3x + 2y = 13 (C)
4x + 2y = 16 (D)
(B) x2
-x = -3
x = 3
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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
y = 2
The solution is x = 3
y = 2
Step 4:
Check answers by substituting into both equations
3x + 2y = 13
2x + y = 8
( = 13)
( = 8)
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Now try Ex 5B
Ch7 (page90 )
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