Simultaneous Equations
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课程笔记
What are Simultaneous Equations?
- Simultaneous equations involve two unknowns (usually x and y) and require two equations to find both values.
- The solution is the pair of values that satisfy both equations at the same time.
- They are called linear if there are no squared terms (like x² or y²).
- Example: 3x + 2y = 11 and 2x – y = 5 have solution x = 3, y = 1.
Solving by Elimination
- Elimination removes one variable by making the coefficients of x (or y) the same in both equations.
- If the signs in front of the term are the same, subtract the equations.
- If the signs are different, add the equations.
- After eliminating one variable, solve the resulting equation for the other variable.
- Substitute the found value into one original equation to find the second variable.
- Always check your solutions in the other original equation.
Solving by Substitution
- Substitution involves rearranging one equation to make x or y the subject (e.g., y = 2x – 5).
- Substitute this expression into the other equation.
- Solve the resulting equation for the remaining variable.
- Substitute back to find the other variable.
- This method is an alternative to elimination.
Solving Graphically
- Plot both equations on the same set of axes (use a table of values or rearrange to y = mx + c).
- The point of intersection of the two lines gives the solution (x, y).
- Example: 2x – y = 3 and 3x + y = 7 intersect at (2, 1), so x = 2, y = 1.
- Graphical method is useful for checking answers but may be less precise.
Solving Graphically
Forming Simultaneous Equations from Word Problems
- Introduce two letters (e.g., x and y) to represent the unknowns, stating what each stands for.
- Translate the given information into two equations.
- Example: '3 apples and 2 bananas cost $1.80' gives 3x + 2y = 180 (in cents).
- Solve the equations simultaneously, then answer the question in context (with units).
- Sometimes you need to find a further value (e.g., product or total cost) after solving.
Worked Example (Elimination)
- Solve: 5x + 2y = 11 and 4x – 3y = 18.
- Multiply first by 3: 15x + 6y = 33. Multiply second by 2: 8x – 6y = 36.
- Add to eliminate y: 23x = 69 → x = 3.
- Substitute x = 3 into 5x + 2y = 11 → 15 + 2y = 11 → 2y = –4 → y = –2.
- Check in second: 4(3) – 3(–2) = 12 + 6 = 18 ✓. Solution: x = 3, y = –2.
Worked Example (Forming Equations)
- Customer 1: 6 bagels + 12 sausage rolls = £9 → 6b + 12s = 9.
- Customer 2: 9 bagels + 10 sausage rolls = £12.30 → 9b + 10s = 12.3.
- Eliminate b: multiply first by 3, second by 2 → 18b + 36s = 27 and 18b + 20s = 24.6.
- Subtract: 16s = 2.4 → s = 0.15. Substitute: 6b + 1.8 = 9 → b = 1.2.
- Cost of 5 bagels and 15 sausage rolls = 5×1.2 + 15×0.15 = 6 + 2.25 = £8.25.
Examiner Tips
- Always check your final solutions satisfy both original equations.
- Write both solutions together (e.g., x = 3, y = –2) to avoid missing one.
- Read the question carefully: sometimes you need to find something else (e.g., product, total cost).
- Show all working clearly, especially when multiplying equations.
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练习题
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1.Solve the simultaneous equations: 3x + 2y = 11 and 2x - y = 5.
Medium- Ax=3, y=1
- Bx=1, y=3
- Cx=2, y=2.5
- Dx=4, y=-0.5
2.Solve the simultaneous equations: 5x - 2y = 26 and 7x + 6y = 10.
Medium- Ax=4, y=-3
- Bx=2, y=-8
- Cx=6, y=2
- Dx=-2, y=-18
3.Solve the simultaneous equations: 6x - 3y = 12 and 2x + 3y = 16.
Medium- Ax=3.5, y=3
- Bx=4, y=4
- Cx=2, y=0
- Dx=1, y=-2
4.Solve the simultaneous equations: 5x + 4y = 10 and 7x - 6y = 43.
Medium- Ax=4, y=-2.5
- Bx=5, y=-3.75
- Cx=3, y=-1.25
- Dx=2, y=0
5.Esme buys x magazines at $2.45 each and y cards at $3.15 each. She spends $60.55 in total and buys 8 magazines. How many cards does she buy?
Medium- A13
- B14
- C12
- D15
6.A shop sells pens and notebooks. A pen costs p cents, a notebook costs n cents. On Monday, 5 pens and 4 notebooks cost 450 cents. On Tuesday, 10 pens and 3 notebooks cost 525 cents. Find the cost of a pen.
Medium- A30 cents
- B45 cents
- C60 cents
- D50 cents
7.Beindu buys 7 apples and 4 bananas for 85 cents, and 3 apples and 8 bananas for 93 cents. Find the cost of an apple (a cents) and a banana (b cents).
Medium- Aa=7, b=9
- Ba=8, b=7
- Ca=9, b=6
- Da=6, b=10
8.A plant costs p dollars and a bush costs b dollars. Ana buys 2 plants and 4 bushes for $42. Paola buys 7 plants and 9 bushes for $107. Find p and b.
Medium- Ap=5, b=8
- Bp=8, b=5
- Cp=6, b=7
- Dp=4, b=9