Simultaneous Equations

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Simultaneous Equations
S5 Int2
Solving Sim. Equations Graphically
Graphs as Mathematical Models
Solving Simple Sim. Equations by Substitution
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Solving Simple Sim. Equations by elimination
Solving harder type Sim. equations
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Starter Questions
S5 Int2
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Simultaneous Equations
S5 Int2
Straight Lines
Learning Intention
Success Criteria

1. To solve simultaneous equations using graphical
   methods.

• Interpret information from a line graph.
• Plot line equations on a graph.
• Find the coordinates where
• 2 lines intersect ( meet)

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Q. Find the equation of each line.
(1,3)
Q. Write down the coordinates where
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
S5 Int2
8cm
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5cm
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Simultaneous Equations
S5 Int2
Straight Lines
Learning Intention
Success Criteria

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

• Draw line graphs given a table of points.
• Find the coordinates where
• 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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Summarise data ! Who should I hire the car from?
Arnold
Total Cost
Up to 2 days Swinton Over 2 days Arnold
Swinton
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
S5 Int2
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Simultaneous Equations
S5 Int2
Straight Lines
Learning Intention
Success Criteria

1. 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 x1
by substitution
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At the point of intersection y coordinates are
equal
2x x1
so we have
2x - x 1
Rearranging we get
x 1
Finally Sub into one of the equations to get y
value
y 2x 2 x 1 2
y x1 1 1 2
OR
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
x1 -x4
so we have
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 1.5 1 2.5
y -x4 -1.5 4 2 .5
OR
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Now try Ex 4 Ch7 (page88 )
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Starter Questions
S5 Int2
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Simultaneous Equations
S5 Int2
Straight Lines
Learning Intention
Success Criteria

1. 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
( 4 10 14)
x 2y 14 x y 9
( 4 5 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
( 14 - 3 11)
2x - y 11 x - y 4
( 7 - 3 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
( 10 - 4 6)
2x - y 6 x y 9
( 5 4 9)
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Now try Ex 5A Ch7 (page89 )
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Starter Questions
S5 Int2
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Simultaneous Equations
S5 Int2
Straight Lines
Learning Intention
Success Criteria

1. 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
(A) x3
2x y 9 x -3y 1
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
( 8 1 9)
2x y 9 x -3y 1
( 4 - 3 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
(A) x1
3x 2y 13 2x y 8
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
( 9 4 13)
3x 2y 13 2x y 8
( 6 2 8)
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Now try Ex 5B Ch7 (page90 )
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