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Simultaneous Equations

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41 views

Simultaneous Equations

Uploaded by

areebhamid28
Copyright
© © All Rights Reserved
Available Formats
Download as PDF, TXT or read online on Scribd
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Simultaneous Equations

What is the difference between an expression and equation?


An expression comprises of coefficients, variables, constants and powers.
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐
An equation has two expressions written on both sides of the ‘=’ sign.
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0
Instances when equations can be solved simultaneously:
1. Line with line situation
1 solution

2. Line with curve situation


1 solution, 2 solutions, no solution

3. Curve with curve situation


1 solution, 2 solutions, no solution
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Methods of solving simultaneous equations:

1. Elimination
• Line with line

2. Substitution
• Line with line
• Line with curve
• Curve with curve

3. Graphical
• Line with line
• Line with curve
• Curve with curve

4. Matrix
• Line with line

Steps of Elimination Method:


1. Multiply both the equations with a constant such that the coefficients of a particular
variable in both the equation becomes same
2. Add or subtract the two equations obtained in step 1 such that a variable is eliminated.
3. Find the value of the variable left.
4. 4. Substitute the value of the variable found in either of the original equations and finds
the value of the other variable.
Steps of Substitution Method:
1. Make the third equation from the easier of the two equation
2. If the third equation is made from the first equation, substitute it in the second equation.
If the third equation is made from the second equation substitutes it in the first equation.
3. Automatically a variable will cancel off and the value of other variable will be found.
4. Insert the value of the variable obtained in the third equation and hence find the value of
other variable.
Steps of Graphical Method:
1. Make y the subject of the formula in both the equations.
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2. Insert some random values of x in the answer of Step 1 and hence find the corresponding
values of y.
3. Plot the values obtained in step 2 on the graph and draw the line or the curve
4. The point where the lines or the curve will intersect is the solution to the simultaneous
equations.

How to distinguish between Linear and Quadratic Equations:


Linear Quadratic
Max Power 1 Max Power 2
If drawn on graph, linear equation results in a If drawn on graph, quadratic equation results
straight line in a curve
1 solution 2 solutions, 1 solution, or no solution
𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐 𝑦𝑦 = 𝑎𝑎𝑥𝑥 2

NOTE: The word ‘intersecting’ means solve simultaneously


Whenever you face a question with variable as power, that question will involve substitution plus
logs (be careful).
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Solve these simultaneous equations. Show your working.

10𝑥𝑥 + 7𝑦𝑦 = −3
5𝑥𝑥 − 𝑦𝑦 = 3
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Solve these simultaneous equations. Show your working.


2𝑥𝑥 + 5𝑦𝑦 = 2
3𝑥𝑥 + 4𝑦𝑦 = −4
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Solve these simultaneous equations. Show your working.


2𝑥𝑥 + 5𝑦𝑦 = 2
3𝑥𝑥 + 4𝑦𝑦 = −4
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Solve these simultaneous equations. Show your working.


2𝑥𝑥 2 + 3𝑦𝑦 2 = 7𝑥𝑥𝑥𝑥
𝑥𝑥 + 𝑦𝑦 = 4
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Type 1 – Basic
Solve these simultaneous equations. Show your working.
𝑥𝑥 + 3𝑦𝑦 = 13
𝑥𝑥 2 + 3𝑦𝑦 2 = 43
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The straight line 3𝑥𝑥 = 2𝑦𝑦 + 18 intersects the curve 2𝑥𝑥 2 − 23𝑥𝑥 + 50 = 0 at the points A and B.
Find the co-ordinates of A and B.
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Type 2 – Fraction
2 3
Solve the simultaneous equations 5𝑥𝑥 + 3𝑦𝑦 = 2 and − =1
𝑥𝑥 𝑦𝑦
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Type 3 – Indices
Solve the simultaneous equations
4𝑥𝑥
= 1024
256𝑦𝑦
32𝑥𝑥 × 9𝑦𝑦 = 243
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Solve, for 𝑥𝑥 and 𝑦𝑦, the simultaneous equations


125𝑥𝑥 = 25(5𝑦𝑦 )
7𝑥𝑥 ÷ 49𝑦𝑦 = 1
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Type 4 – Word Problem


A B
P

D C
The diagram shows a square ABCD of area 60 m2 . The point P lies on BC and the sum of the
lengths AP and BP is 12 m. Given that the lengths of AP and BP are x m and y m respectively,
form two equations in x and y and hence find the length of BP.
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Type 5 – Substitution
Solve
1 1
3𝑥𝑥 2 − 𝑦𝑦 −2 = 4
1 1
4𝑥𝑥 2 + 3𝑦𝑦 −2 = 14
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Type 6 – Finding Unknowns


The graph of the curve 𝑦𝑦 = 𝑝𝑝(42𝑥𝑥 ) − 𝑞𝑞(4𝑥𝑥 ) passes through the points (0, 2) and (0.5, 14). Find
the value of p and of q.
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Practice Questions

Q1 May June 2004 P1 Q2

Q2 May June 2004 P1 Q5

Q3 May June 2005 P2 Q7

Q4 October November 2005 P2 Q2

Q5 May June 2006 P1 Q2


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Q6 May June 2006 P1 Q11

Q7 October November 2006 P2 Q5

Q8 May June 2007 P1 Q3

Q9 May June 2008 P2 Q3

Q10 October November 2009 P1 Q4

Q11 May June 2010 P12 Q1


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Q12 October November 2011 P12 Q2

Q13 May June 2012 P12 Q4

Q14 May June 2013 P12 Q5

Q15 October November 2013 P22 Q5

Q16 May June 2014 P22 Q8

Q17 October November 2014 P22 Q6

Q18 May June 2015 P22 Q5


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Q1

Q2

Q3

Q4
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Q5

Q6

Q7
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Q8

Q9

Q10
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Q11

Q12

Q13
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Q14

Q15
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Q16

Q17
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Q18
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