Transcript Simultaneous Equations

Chapter 17
Simultaneous Equations
Learning Objectives
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Use graphs to solve simultaneous equations
Show that certain simultaneous equations
have no solution
Solve simultaneous equation by elimination
Solve simultaneous equations by substitution
Solve practical problems with simultaneous
equations
Using Graphs to Solve
Simultaneous Equations
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Pick 3 x-points for each of the
equations
Work out the y-values
Draw each of the equations
5
4
3
Examples
1. Solve
x + 2y = 5
x – 2y = 1
2
1
0
-1
0
1
2
3
4
5
Using Graphs to Solve
Simultaneous Equations
5
2. Solve
x + 2y = 5
x – 2y = 1
4
3
2
1
0
-1
0
-1
1
2
3
4
5
Using Graphs to Show Simultaneous Equations
Can Have NO Solution
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If the two graphs do not cross each other
there is no solution
5
4
Examples
1. Show that
y – 2x = 4
2y = 4x – 1
have no solution
3
2
1
0
-1
0
-1
1
2
3
4
5
Using Algebra to Show that Simultaneous
Equations Can Have NO Solution
If the two graphs have the same gradient
then they are parallel and do not cross
so they do not have a solution
 Re-arrange the equation to put it in the
form y = mx + c (remember that m =
grad)
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Examples
1.
Show that y – 2x = 4 and 2y = 4x – 1
do not have any solution.
2.
Show that y + 1 = -5x and 2y + 10x = 5
do not have any solution
The Elimination Method
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Add or subtract the equation to eliminate
either the x or the y
Examples
1. Solve 2x + 3y = 9 and 2x + y = 7
2.
Solve x + 2y = 5 and x – 2y = 1
3.
Solve 2x – y = 1 and 3x + y = 9
Harder Elimination
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If neither the x’s nor the y’s are the same
make them the same by multiplying either 1
or 2 of the equations
Examples
1. 5x + 2y = 11 and 3x – 4y = 4
2.
3x + 7y = -2 and 4x + 9 = -3y
3.
9x = 4y – 20 and 5x = 6y - 13
The Substitution Method
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Re-arrange so that one of the equations has
either x= or y=
Then substitute this into the other equation
Examples
1. Solve 5x + y = 9 and y = 4x
2.
Solve x + 4y = 32 and x = 2y - 4
Practical Problems
Examples
1. Billy buys 5 1st class stamps and 3 2nd
class stamps for £1.93.
Jane buys 3 1st class and 5 2nd class
stamps for £1.83.
How much is a 1st class and 2nd class
stamp?
Practical Problems
2. Micro-scooters costs £x each and pogo
sticks cost £y each.
2 micro-scooters and 4 pogo sticks cost
£65
1 micro-scooter and 3 pogo sticks cost
£40
Find the value of x and y