IBMaths: AA HLDPTopic Questions1. Number & Algebra1.10 Systems of Linear Equations

Systems of Linear Equations (DP IB Maths: AA HL)

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1.10 Systems of Linear Equations

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1a
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Solve the following simultaneous equations. 

5 x minus 3 y equals 19 

2 x plus y equals 1

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1b
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a minus 11 b equals 23 

5 a plus 5 b equals negative 5

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1c
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5 over 4 space m minus 3 over 2 space n equals negative 9 over 8  

1 half space m plus 5 over 3 space n equals 11 over 36

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2
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Use the method of substitution to solve the following systems of linear equations.

(i)


x minus y minus z equals 0

 2 x plus y minus 3 z equals 5 

2 x minus 3 y plus 4 z equals 4 

(ii)


2 x minus y minus 3 z equals 3 

3 x plus 2 y minus 2 z equals 12 

2 x plus y plus 2 z equals negative 7

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3
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A festival charges $ x USD for an adult ticket, $ y USD for a child ticket and $ zUSD for a car parking pass.

Given that 4 adult tickets, 7 child tickets and 2 car passes cost $540 USD, 2 adult tickets, 2 child tickets and 1 car pass cost $210 USD and 7 adult tickets and 3 car passes cost $450 USD, 

(i)
set up a system of linear equations in three unknowns,
 
(ii)
find the values of x, y, and z.

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4
Sme Calculator6 marks

Solve the following system of linear equations.  

3 x plus 2 y minus z equals 1 

x minus y plus 5 z equals negative 2 

2 x plus y equals 3

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5
Sme Calculator6 marks

Solve the following the system of linear equations. 

2 x plus 2 y minus 3 z equals negative 8 

3 x plus 2 y minus z equals 0 

x minus y plus z equals 11

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6a
Sme Calculator4 marks

Consider the system of equations 

negative 6 a plus left parenthesis k minus 3 right parenthesis b equals 1 

3 k a minus 5 b equals 4 

Find the values of the real parameter k such that the system has a unique solution.

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6b
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Find the unique solution in terms of k.

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7
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Solve the following system of equations using row operations. 

3 x plus 9 y minus 3 z equals 45 

6 x plus 3 y plus 3 z equals 21 

3 x minus 3 y minus 6 z equals 0

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8
Sme Calculator6 marks

Consider the following system of equations 

2 x plus y minus 3 z equals negative 4 

x minus y plus 2 z equals 2 

4 x plus 2 y minus 6 z equals k

where k element of straight real numbers 

Show that the system has no unique solution for any value of k.

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1a
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Solve the following simultaneous equations. 

1 half x plus y equals 9

x minus 2 y equals 2

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1b
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3 a minus 5 b equals 30

5 a plus 2 b equals 3.5

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1c
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fraction numerator 5 m over denominator 6 end fraction minus fraction numerator 3 n over denominator 4 end fraction equals 2 n

fraction numerator 2 m over denominator 3 end fraction minus fraction numerator 4 n over denominator 5 end fraction equals negative 2 over 3

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2
Sme Calculator6 marks

Use an algebraic method to solve the following system of linear equations.

2 x minus y plus 3 z equals 4

3 x plus 2 y plus 6 z equals negative 5

2 x minus 4 y minus z equals 8

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3
Sme Calculator6 marks

Use an algebraic method to solve the following system of linear equations.

2 x plus 3 y plus 4 z equals 15

x minus 2 y minus 6 z equals negative 5

2 x minus 6 y minus 5 z equals 6

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4
Sme Calculator6 marks

Use an algebraic method to solve the following system of linear equations.

2 x plus 2 y minus 3 z equals 8

3 x minus y plus 2 z equals 6

x minus 2 y plus 4 z equals negative 3

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5
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Two straight lines have equations y equals 3 over 2 x plus 1 and 4 x minus 3 y minus 1 equals 0. Find the coordinates of their point of intersection.

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6
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Consider the polynomial f open parentheses x close parentheses equals x to the power of 4 plus a x cubed plus b x squared plus c x plus 12 

Given that f open parentheses 1 close parentheses equals f open parentheses 2 close parentheses equals f open parentheses 3 close parentheses equals 0 comma 

(i)
set up a system of linear equations in three unknowns,
(ii)
hence, find the values of a comma space b comma and c.

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7
Sme Calculator6 marks

Consider the following system of linear equations.

 2 x plus y minus 2 z equals 6

2 x minus 2 y plus 3 z equals negative 5

negative 2 x plus 3 y plus a z equals b 

Given that the system has no solutions, find the value of a and the set of possible values of b.

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8a
Sme Calculator6 marks

Consider the following system of equations

a x plus y plus z equals 4

x plus y plus z equals a

x minus y plus a z equals 2

Find, in terms of a, expressions for x comma space y and z.

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8b
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Find value(s) of the real parameter a so that the system has no unique solutions.

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8c
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Given a equals 0, find the values of x comma space y spaceand z.

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9
Sme Calculator8 marks

The following system of equations has an infinite number of solutions

 x minus 2 y plus z equals k

x plus y minus z equals 2

3 x minus 3 y plus z equals 12

(i)
Find the value of k,
(ii)
Find the general solution.

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10a
Sme Calculator4 marks

Consider the system of equations

 x plus k y plus z equals k

x plus 2 y plus 3 z equals 0

3 x plus 8 y plus 5 z equals 6 

When k equals m, the system does not have a unique solution. Find the value of m.

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10b
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Given that k not equal to m, show that the solution to the system is independent of k and hence find the unique solution.

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1a
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Solve the following simultaneous equations.

5 x minus 2 y equals 9.5
negative 2 x minus 5 y equals 16.5

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1b
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3 open parentheses a minus 2 b close parentheses equals 7 minus b
2 open parentheses 2 a minus b close parentheses equals 5 b minus 11

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1c
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fraction numerator 2 open parentheses p minus q close parentheses over denominator 5 end fraction minus fraction numerator 3 p over denominator 2 end fraction equals fraction numerator p minus 4 q over denominator 10 end fraction

fraction numerator 2 p over denominator 3 end fraction minus fraction numerator 4 q over denominator 5 end fraction equals 1 minus 2 over 3 q

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2
Sme Calculator6 marks

Use an algebraic method to solve the following system of linear equations.

2 x minus 3 y minus 5 z equals 4

x minus 4 y plus 6 z equals negative 6

3 y minus 2 x minus 3 z equals 0

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3
Sme Calculator1 mark

Solve the following system of linear equations using an algebraic method.

2 x minus 3 y plus 4 z equals negative 1

x minus 4 y minus 6 z equals 8

5 y minus 3 x minus 5 z equals 1

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4
Sme Calculator6 marks

Use an algebraic method to solve the following system of linear equations.

3 y minus 2 x plus 5 z equals 14

3 x minus 2 y minus 2 z equals 11

3 z minus 4 x minus 4 y equals 35

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5
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Find the unique point of intersection of the planes with the following Cartesian equations:

x minus 3 y plus 2 z equals negative 3

z plus 2 y minus x equals 4

3 z plus y minus 4 x equals 1

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6
Sme Calculator6 marks

Consider the function f defined by f open parentheses x close parentheses equals x to the power of 4 plus a x cubed plus b x squared plus c x plus 24. The graph y equals f open parentheses x close parenthesespasses through the points open parentheses 1 comma 36 close parentheses comma space open parentheses 2 comma 24 close parentheses and open parentheses 3 comma 0 close parentheses. 

Find the values of a comma space b spaceand c.

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7a
Sme Calculator5 marks

Consider the system of equations

2 a plus 6 b plus x c equals y

6 b minus a minus c equals 7

a plus 2 b minus 3 c equals 1

where x and y are real constants.

Find the value of x such that the system does not have a unique solution.

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7b
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Given that c equals 0 find the values of a comma space b and y such that the system of equations has a solution

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8a
Sme Calculator3 marks

Consider the following system of equations

x plus 2 y minus z equals 0

2 x minus y plus 4 z equals 0

x minus 3 y plus open parentheses 1 minus a close parentheses z equals 4 minus a squared

Show that the system has a unique solution when a not equal to negative 4.

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8b
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For the case where a not equal to negative 4, state the solution in terms of a.

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8c
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For the case where a equals negative 4, show that there are no solutions.

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9a
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The following system of equations has an infinite number of solutions.

3 x minus 2 y minus 7 z equals 3

2 x plus y minus 4 z equals 1

3 y minus x plus 3 z equals k

Find the value of k.

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9b
Sme Calculator4 marks

Find the general solution.

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10a
Sme Calculator3 marks

Consider the following system of linear equations.

x minus 3 y plus 3 z equals k

3 x minus 2 y plus z equals 4

2 x plus y minus 2 z equals 2

Show that the system of equations does not have a unique solution.

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10b
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Find the value of k for which the equations are consistent.

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10c
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For the value of k found in part (b), find the general solution of these equations.

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11
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The rate, R, of increase of the volume of a cloud created in a science lab is related to the change in air temperature,T , and air pressure, P, by the equation

 R equals k T to the power of x P to the power of y, where x comma space y comma space k space element of space straight real numbers.

A meteorologist takes measurements at three intervals and records the data as follows.

Measurement

  bold italic R bold space stretchy left parenthesis c m cubed s to the power of negative 1 end exponent stretchy right parenthesis  bold italic T bold space stretchy left parenthesis degree C stretchy right parenthesis  bold italic P begin bold style stretchy left parenthesis kPa stretchy right parenthesis end style

1

48.75

17.1

101.2

2

46.13

15.9

101.8

3

43.47

14.7

102.5

Find x comma space y spaceand k.

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