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Simultaneous Equations
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S5 Int2
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
4-Jul-17
Starter Questions
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S5 Int2
1
% in one year.
3
If its initial price was £12 000. How much is it now.
1. A car reduced in value by 33
2. Tidy up the expression
-6e + 5d - 7d + 6e
3.
Calculate
a)
13y
- 5y
4-Jul-17
b)
12x
+ 4x
Simultaneous Equations
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S5 Int2
Straight Lines
Learning Intention
1. To solve simultaneous
equations using graphical
methods.
Success Criteria
1. Interpret information from a
line graph.
2. Plot line equations on a graph.
3. Find the coordinates were 2
lines intersect ( meet)
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
4
Q. Find the equation
of each line.
Red : y  x  2
2
Blue : y  2x  1
Q. Write down the
coordinates were
they meet.
1
2
1
0
1
2
3
4
4-Jul-17
(1,3)
3
1
2
3
4
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S5 Int2
Simultaneous Equations
Straight Lines
2
Q. Find the equation
of each line.
1.5
Red : y  x
1
Blue : y   x  1
Q. Write down the
coordinates
where they meet.
0.5
2
1.5
1
(-0.5,-0.5)
0.5
0
0.5
1
1.5
2
4-Jul-17
0.5
1
1.5
2
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S5 Int2
Simultaneous Equations
Straight Lines
2
Q. Plot the lines.
1.5
y  x
Blue : y  2x  1
Q. Write down the
coordinates
where they meet.
0.5
2
1.5
1
0.5
0
0.5
1
1.5
2
4-Jul-17
(1,1)
1
0.5
1
1.5
2
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S5 Int2
Simultaneous Equations
Straight Lines
Now try Exercise 2
Ch7 (page 84 )
4-Jul-17
Starter Questions
S5 Int2
1
% in one year.
2
If its initial price was £96 000. How much is it now.
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1. A house has increased in value by 12
2.
Find the area of the shape
8cm
3.
4-Jul-17
Calculate
55% of 360
5cm
Simultaneous Equations
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S5 Int2
Straight Lines
Learning Intention
1. To use graphical methods
to solve real-life
mathematical models
4-Jul-17
Success Criteria
1. Draw line graphs given a table
of points.
2. Find the coordinates were 2
lines intersect ( meet)
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S5 Int2
Simultaneous Equations
Straight Lines
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.
Anrnold Palmer Car Hire
Per Day
0
1
2
3
4
5
Total Cost £
£100
120
140
160
180
200
Swinton Direct Car Hire
Per day
0
1
2
3
4
5
Total Cost £
0
60
120
180
240
300
4-Jul-17
Straight Lines
Summarise data !
Who should I hire
the car from?
Total Cost £
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S5 Int2
Simultaneous Equations
4-Jul-17
Up to 2 days
Swinton
Over 2 days Arnold
Days
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S5 Int2
Simultaneous Equations
Straight Lines
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.
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
Now try Exercise 3
Ch7 (page 85 )
4-Jul-17
Starter Questions
S5 Int2
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1. Write down the gradient and were line crosses y - axis.
y = -2x + 6
2. Rearrange the equations below into y =
3.
(a)
y + 2x = 8
(b)
4x + 2y =10
Calculate
a)
-5x
+ 5x
4-Jul-17
b)
-19y
- 9y
Simultaneous Equations
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S5 Int2
Straight Lines
Learning Intention
1. To solve pairs of equations
by substitution.
4-Jul-17
Success Criteria
1. Apply the process of
substitution to solve simple
simultaneous equations.
S5 Int2
Simultaneous Equations
Straight Lines
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Example 1
3
Solve the equations
2
y = 2x
y = x+1
1
3
by substitution
2
0
1
1
2
3
4-Jul-17
1
2
3
S5 Int2
Simultaneous Equations
Straight Lines
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At the point of intersection y coordinates are equal:
y = 2x
y = x+1
2x = x+1
so we have
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
The solution is x = 1
4-Jul-17
y = x+1 = 1 + 1 = 2
y = 2 or (1,2)
S5 Int2
Simultaneous Equations
Straight Lines
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Example 1
Solve the equations
4
3.5
(1.5, 2.5)
3
y = x + 1
x + y = 4
by substitution
2.5
2
1.5
1
0.5
0
4-Jul-17
0.5
1
1.5
2
2.5
3
3.5
4
The solution is x = 1.5 y = 2.5 (1.5,2.5)
S5 Int2
Simultaneous Equations
Straight Lines
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At the point of intersection y coordinates are equal:
y = x +1
y =-x+ 4
x+1
Rearranging we get : 2x
2x
x
so we have
=
=
=
=
-x+4
4 - 1
3
3 ÷ 2 = 1.5
Finally : Sub into one of the equations to get y value
y = x +1 = 1.5 + 1 = 2.5
OR
4-Jul-17
y = -x+4 = -1.5 + 4 = 2 .5
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S5 Int2
Simultaneous Equations
Straight Lines
Now try Ex 4
Ch7 (page88 )
4-Jul-17
Starter Questions
S5 Int2
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1. A house appreciates by 20% in one year.
If its initial price was £60 000. How much is it now.
2. Rearrange the equations below into y =
3.
(a)
y + x=2
(b)
x + y =5
Calculate
a)
5y
- 2y
4-Jul-17
b)
12w
+ 9w
Simultaneous Equations
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S5 Int2
Straight Lines
Learning Intention
1. To solve simultaneous
equations of 2 variables.
Success Criteria
1. Understand the term
simultaneous equation.
2. Understand the process for
solving simultaneous
equation of two variables.
3. Solve simple equations
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
Example 1
Solve the equations
x + 2y = 14
x + y = 9
by elimination
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
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
4-Jul-17
S5 Int2
Simultaneous Equations
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
x = 9 - 5
x = 4
Step 4:
x + 2y = 14
x + y = 9
4-Jul-17
The solution is x = 4
y = 5
Check answers by substituting into both equations
( 4 + 10 = 14)
( 4 + 5 = 9)
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S5 Int2
Simultaneous Equations
Straight Lines
Example 2
Solve the equations
2x - y = 11
x - y = 4
by elimination
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
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
4-Jul-17
S5 Int2
Simultaneous Equations
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
y = 7 - 4
y = 3
Step 4:
2x - y = 11
x - y = 4
4-Jul-17
The solution is x =7
y =3
Check answers by substituting into both equations
( 14 - 3 = 11)
( 7 - 3 = 4)
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S5 Int2
Simultaneous Equations
Straight Lines
Example 3
Solve the equations
2x - y = 6
x + y = 9
by elimination
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
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
x + y = 9
(A)
(B)
3x = 15
4-Jul-17
x = 15 ÷ 3 = 5
S5 Int2
Simultaneous Equations
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
y = 9 - 5
Step 4:
y = 4
The solution is x = 5
y = 4
Check answers by substituting into both equations
2x - y = 6
x + y = 9
4-Jul-17
( 10 - 4 = 6)
( 5 + 4 = 9)
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S5 Int2
Simultaneous Equations
Straight Lines
Now try Ex 5A
Ch7 (page89 )
4-Jul-17
Starter Questions
S5 Int2
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1. A fridge is increased by 4%.
If its old price was £240. Find the new price.
2. Tidy up the expressions
3.
(a)
2y - x - 3y + 2x
(b)
4x - 5y + 6x + 8y
Solve the equations :
x + y=6
x- y= 4
4-Jul-17
Simultaneous Equations
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S5 Int2
Straight Lines
Learning Intention
1. To solve harder
simultaneous equations of
2 variables.
4-Jul-17
Success Criteria
1. Apply the process for solving
simultaneous equations to harder
examples.
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S5 Int2
Simultaneous Equations
Straight Lines
Example 1
Solve the equations
2x + y = 9
x - 3y = 1
by elimination
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
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 (A) x3
x -3y = 1 (B) x1
4-Jul-17
6x + 3y = 27 (C)
x - 3y = 1 (D)
7x = 28
x = 28 ÷ 7 = 4
S5 Int2
Simultaneous Equations
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
y = 1
Step 4:
The solution is x = 4
y = 1
Check answers by substituting into both equations
2x + y = 9
x -3y = 1
4-Jul-17
( 8 + 1 = 9)
( 4 - 3 = 1)
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S5 Int2
Simultaneous Equations
Straight Lines
Example 2
Solve the equations
3x + 2y = 13
2x + y = 8
by elimination
4-Jul-17
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S5 Int2
Simultaneous Equations
Straight Lines
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 (A) x1
2x + y = 8 (B) x2
4-Jul-17
3x + 2y = 13 (C)
4x + 2y = 16 (D)
-x = -3
x = 3
S5 Int2
Simultaneous Equations
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
y = 2
Step 4:
The solution is x = 3
y = 2
Check answers by substituting into both equations
3x + 2y = 13
2x + y = 8
4-Jul-17
( 9 + 4 = 13)
( 6 + 2 = 8)
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S5 Int2
Simultaneous Equations
Straight Lines
Now try Ex 5B
Ch7 (page90 )
4-Jul-17