IB DP Maths: AA HL復習筆記1.10.1 Systems of Linear Equations
What are systems of linear equations?
- A linear equation is an equation of the first order (degree 1)
- This means that the?maximum degree?of each term is 1
- These are examples of linear equations:
- 2x?+ 3y?= 5 &?5x?–?y?= 10 + 5z
- These are examples of non-linear equations:
- x2?+?5x?+ 3 = 0 &?3x?+ 2xy?– 5y?= 0
- The terms?x2?and?xy?have degree 2
- A system of linear equations is where?two or more linear equations?work together
- These are also called?simultaneous equations
- If there are?n?variables?then you will need?at least?n?equations?in order to solve it
- For your exam?n?will be 2 or 3
What do systems of linear equations represent?
- The most common application of systems of linear equations is in?geometry
- For a?2×2?system
- Each equation will represent a?straight line in 2D
- The solution (if it exists and is unique) will correspond to the?coordinates?of the point where the?two lines intersect
- For a?3×3?system
- Each equation will represent a?plane in 3D
- The solution (if it exists and is unique) will correspond to the?coordinates?of the point where the?three planes intersect
Systems of Linear Equations
How do I set up a system of linear equations?
- Not all questions will have the equations written out for you
- There will be?bits of information?given about the variables
- Two bits?of information for a?2×2 system
- Three bits?of information for a?3×3 system
- Look out for clues such as ‘assuming a linear relationship’
- Choose to assign?x,?y?&?zto the given variables
- This will be helpful if using a GDC to solve
- Or you can choose to use more meaningful variables if you prefer
- Such as?c?for the number of cats and?d?for the number of dogs
How do I use my GDC to solve a system of linear equations?
- You can use your?GDC to solve?the system on the?calculator papers (paper 2 & paper 3)
- Your GDC will have a function within the algebra menu to solve a system of linear equations
- You will need to choose the number of equations
- For two equations the variables will be?x?and?y
- For three equations the variables will be?x, y?and?z
- If required, write the equations in the given form
- ax?+?by?=?c
- ax?+?by?+?cz?=?d
- Your GDC will display the values of?x?and?y?(or?x, y,?and?z)
Exam Tip
- Make sure that you are familiar with how to use your GDC to solve a system of linear equations because even if you are asked to use an algebraic method and show your working, you can use your GDC to check your final answer
- If a systems of linear equations question is asked on a non-calculator paper, make sure you check your final answer by inputting the values into all original equations to ensure that they satisfy the equations
Worked Example
On a mobile phone game, a player can purchase one of three power-ups (fire, ice, electricity) using their points.
- Adam buys 5 fire, 3 ice and 2 electricity power-ups costing a total of 1275 points.
- Alice buys 2 fire, 1 ice and 7 electricity power-ups costing a total of 1795 points.
- Alex buys 1 fire and 1 ice power-ups which in total costs 5 points less than a single electricity power up.
Find the cost of each power-up.
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