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Solving Sim. Equations Graphically
Solving Simple Sim. Equations by Substitution
Simultaneous Simultaneous EquationsEquations
Solving Simple Sim. Equations by elimination
Solving harder type Sim. equations
Graphs as Mathematical Models
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Starter QuestionsStarter Questions
2. T
11. A car reduced in value by 33 % in one year.
3 I f its initial price was £12 000. How much is it now.
1.1. Apply the process of Apply the process of substitution to solve simple substitution to solve simple simultaneous equations.simultaneous equations.
Straight LinesS5
Int2
Simultaneous EquationsStraight Lines
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Example 1
Solve the equations
y = 2x y = x+1
by substitution
3 2 1 0 1 2 3
3
2
1
1
2
3
Simultaneous EquationsStraight Lines
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At the point of intersection y coordinates are equal:
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 = 1 + 1 = 2
so we have
y = 2x y = x+1
The solution is x = 1 y = 2 or (1,2)
Simultaneous EquationsStraight Lines
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Example 1
Solve the equations
y = x + 1 x + y = 4
by substitution
0 0.5 1 1.5 2 2.5 3 3.5 4
0.5
1
1.5
2
2.5
3
3.5
4
(1.5, 2.5)
Simultaneous EquationsStraight Lines
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At the point of intersection y coordinates are equal:
x+1 = -x+4 Rearranging we get : 2x = 4 - 1
2x = 3
Finally : Sub into one of the equations to get y value
y = x +1 = 1.5 + 1 = 2.5
y = -x+4 = -1.5 + 4 = 2 .5
so we have
y = x +1y =-x+ 4
The solution is x = 1.5 y = 2.5 (1.5,2.5)
x = 3 ÷ 2 = 1.5
OR
Simultaneous EquationsStraight Lines
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Now try Ex 4Ch7 (page88 )
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Starter QuestionsStarter Questions
2. Rearrange the equations below into y =
1. A house appreciates by 20% in one year.
I f its initial price was £60 000. How much is it now.
1. To solve simultaneous equations of 2 variables.
Simultaneous Equations
1.1. Understand the term Understand the term simultaneous equation.simultaneous equation.
2.2. Understand the process Understand the process for solving simultaneous for solving simultaneous equation of two variables.equation of two variables.
3.3. Solve simple equationsSolve simple equations
Straight Lines
Simultaneous EquationsStraight Lines
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Example 1
Solve the equations
x + 2y = 14 x + y = 9
by elimination
Simultaneous EquationsStraight Lines
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Step 1: Label the equationsx + 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
Simultaneous EquationsStraight Lines
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Step 3: Sub into one of the equations to get other variable
Substitute y = 5 in (2)
x + y = 9 (2)
x + 5 = 9
The solution is x = 4 y = 5
Step 4: Check answers by substituting into both equations
x = 9 - 5
x = 4
x + 2y = 14x + y = 9
( 4 + 10 = 14)( 4 + 5 = 9)
Simultaneous EquationsStraight Lines
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Example 2
Solve the equations
2x - y = 11 x - y = 4
by elimination
Simultaneous EquationsStraight Lines
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Step 1: Label the equations2x - 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
Simultaneous EquationsStraight Lines
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Step 3: Sub into one of the equations to get other variable
Substitute x = 7 in (2)
x - y = 4 (2)
7 - y = 4
The solution is x =7 y =3
Step 4: Check answers by substituting into both equations
y = 7 - 4
y = 3
2x - y = 11 x - y = 4
( 14 - 3 = 11)( 7 - 3 = 4)
Simultaneous EquationsStraight Lines
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Example 3
Solve the equations
2x - y = 6 x + y = 9
by elimination
Simultaneous EquationsStraight Lines
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Step 1: Label the equations2x - 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
Simultaneous EquationsStraight Lines
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Step 3: Sub into one of the equations to get other variable