1. Simultaneous Equations
Solving Sim. Equations Graphically
Graphs as Mathematical Models
Solving Simple Sim. Equations by Substitution
Solving Simple Sim. Equations by elimination
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2. Starter Questions
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3. Simultaneous Equations
S5 Int2
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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4. (1,3) Q. Find the equation of each line. Q. Write down the
coordinates were
they meet.
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5. Q. Find the equation of each line. Q. Write down the coordinates
where they meet.
(-0.5,-0.5)
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6. Q. Plot the lines. (1,1) Q. Write down the coordinates
where they meet.
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7. Starter Questions
S5 Int2
8cm
5cm
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8. Simultaneous Equations
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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9. 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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10. Who should I hire the car from?
Summarise data !
Who should I hire the car from?
Total Cost £
Arnold
Up to 2 days Swinton
Over 2 days Arnold
Swinton
Days
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11. 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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12. Starter Questions
S5 Int2
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13. Simultaneous Equations
S5 Int2
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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14. Example 1
Solve the equations
y = 2x
y = x+1
by substitution
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15. 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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16. y = x + 1 x + y = 4 by substitution (1.5, 2.5) Example 1
Solve the equations
y = x + 1
x + y = 4
by substitution
(1.5, 2.5)
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17. 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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18. Starter Questions
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19. Simultaneous Equations
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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20. Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination
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21. Step 1: Label the equations
x + 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
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22. Substitute y = 5 in (2) x + y = 9 (2) x + 5 = 9 x = 9 - 5
Step 3: Sub into one of the equations to get other variable
Substitute y = 5 in (2)
x + y = 9 (2)
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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23. Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination
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24. Step 1: Label the equations
2x - 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
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25. Substitute x = 7 in (2) x - y = 4 (2) 7 - y = 4 y = 7 - 4
Step 3: Sub into one of the equations to get other variable
Substitute x = 7 in (2)
x - y = 4 (2)
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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26. Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination
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27. Step 1: Label the equations
2x - 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
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28. Substitute x = 5 in (2) x + y = 9 (2) 5 + y = 9 y = 9 - 5
Step 3: Sub into one of the equations to get other variable
Substitute x = 5 in (2)
x + y = 9 (2)
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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29. Starter Questions
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30. Simultaneous Equations
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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31. Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination
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32. Step 1: Label the equations
2x + y = 9 (1)
x -3y = 1 (2)
Step 2: Decide what you want to eliminate
Adding
Eliminate y by :
2x + y = 9
x -3y = 1
(1) x3
6x + 3y = 27 (3)
x - 3y = 1 (4)
(2) x1
7x = 28
x = 28 ÷ 7 = 4
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33. Substitute x = 4 in equation (1)
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
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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