IB Mathematics AA Systems of Liner Equations Study Notes
IB Mathematics AA Systems of Liner Equations Study Notes
IB Mathematics AA Systems of Liner Equations Study Notes Offer a clear explanation of Systems of Liner Equations , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Systems of Liner Equations.
Systems of Linear Equations
The study of systems of linear equations is fundamental to understanding mathematical relationships and solving real-world problems. This topic focuses on solving systems of linear equations, including unique solutions, infinite solutions, and inconsistent systems. The syllabus encourages the use of both algebraic and technological methods, such as row reduction or matrix operations.
Content Guidance, Clarification, and Syllabus Links:
- Solutions of Systems of Linear Equations:
Systems of up to three equations in three unknowns should be solved using both algebraic and technological methods. Common methods include:
- Algebraic Methods: Substitution, elimination, and Gaussian elimination.
- Technological Methods: Matrix operations (row reduction) or computational tools like graphing calculators and software.
- Types of Solutions:
Systems of linear equations can result in:
- Unique Solution: The system has a single point of intersection.
- Infinite Solutions: The system has infinitely many solutions, typically represented by a general solution.
- No Solution: The system is inconsistent and the equations represent parallel lines or non-intersecting planes.
- Infinite Solutions:
For systems with an infinite number of solutions, a general solution is provided, typically in parametric form.
- Connections:
- Linked to the intersection of lines and planes, exploring cases in three-dimensional space (AHL 3.18).
- Use of matrices for solving systems, including row reduction and reduced row echelon form (RREF).
Theory of Knowledge (TOK) Connections:
- Mathematics, Sense, Perception, and Reason: Can solutions in higher dimensions lead us to reason that these spaces exist beyond our sense perception?
Examples:
- Example 1: Solve the system of equations:
- \( x + y + z = 6 \)
- \( 2x – y + 3z = 14 \)
- \( -x + 2y – z = -2 \)
Solution: Use substitution, elimination, or matrix row reduction to find \( x = 2, y = 1, z = 3 \).
- Example 2: Analyze the system:
- \( x + 2y – z = 4 \)
- \( 3x + 6y – 3z = 12 \)
- \( -x – 2y + z = -4 \)
Observation: This system has infinitely many solutions since the second and third equations are scalar multiples of the first. The general solution can be represented parametrically.
- Example 3: Consider the system:
- \( x + y + z = 2 \)
- \( 2x + 2y + 2z = 6 \)
- \( x – y + z = 1 \)
Observation: This system has no solutions as it is inconsistent (contradictory equations).
Exploration Questions:
- How does the use of matrices simplify solving complex systems of equations?
- What are the real-world applications of solving systems of linear equations?
IB Mathematics AA SL Transformation of Graphs Exam Style Worked Out Questions
Question
The following diagram shows the graph of a function \(f\), for −4 ≤ x ≤ 2.
On the same axes, sketch the graph of \(f\left( { – x} \right)\).
Another function, \(g\), can be written in the form \(g\left( x \right) = a \times f\left( {x + b} \right)\). The following diagram shows the graph of \(g\).
Write down the value of a and of b.
▶️Answer/Explanation
Markscheme
A2 N2
[2 marks]
recognizing horizontal shift/translation of 1 unit (M1)
eg b = 1, moved 1 right
recognizing vertical stretch/dilation with scale factor 2 (M1)
eg a = 2, y ×(−2)
a = −2, b = −1 A1A1 N2N2
[4 marks]
Question
Let \(f(x) = 3{(x + 1)^2} – 12\) .
Show that \(f(x) = 3{x^2} + 6x – 9\) .[2]
For the graph of f
(i) write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the y-intercept;
(iv) find both x-intercepts.[8]
Hence sketch the graph of f .[2]
Let \(g(x) = {x^2}\) . The graph of f may be obtained from the graph of g by the two transformations:
a stretch of scale factor t in the y-direction
followed by a translation of \(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right)\) .
Find \(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right)\) and the value of t.[3]
▶️Answer/Explanation
Markscheme
\(f(x) = 3({x^2} + 2x + 1) – 12\) A1
\( = 3{x^2} + 6x + 3 – 12\) A1
\( = 3{x^2} + 6x – 9\) AG N0
[2 marks]
(i) vertex is \(( – 1{\text{, }} – 12)\) A1A1 N2
(ii) \(x = – 1\) (must be an equation) A1 N1
(iii) \((0{\text{, }} – 9)\) A1 N1
(iv) evidence of solving \(f(x) = 0\) (M1)
e.g. factorizing, formula,
correct working A1
e.g. \(3(x + 3)(x – 1) = 0\) , \(x = \frac{{ – 6 \pm \sqrt {36 + 108} }}{6}\)
\(( – 3{\text{, }}0)\), \((1{\text{, }}0)\) A1A1 N1N1
[8 marks]
A1A1 N2
Note: Award A1 for a parabola opening upward, A1 for vertex and intercepts in approximately correct positions.
[2 marks]
\(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{ – 1}\\
{ – 12}
\end{array}} \right)\) , \(t = 3\) (accept \(p = – 1\) , \(q = – 12\) , \(t = 3\) ) A1A1A1 N3
[3 marks]