Simultaneous Equations

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Linear Simultaneous Equations

Linear simultaneous equations involve two unknowns (usually x and y) and two equations, e.g. 3x + 2y = 11 and 2x - y = 5.
The solution is the pair of values (x, y) that satisfy both equations simultaneously.
Solve by elimination: make the coefficients of one variable the same, then add or subtract to eliminate it.
If the signs in front of the term to eliminate are the same, subtract the equations.
If the signs are different, add the equations.
Solve the resulting equation for one variable, then substitute back to find the other.
Solve by substitution: rearrange one equation to y = ... (or x = ...) and substitute into the other equation.
Always check your final solutions by substituting into both original equations.

Graphical Solution of Linear Simultaneous Equations

Plot both equations on the same set of axes (use a table of values or rearrange to y = mx + c).
The point of intersection gives the solution: x-coordinate is the x-value, y-coordinate is the y-value.
If the lines are parallel, there is no solution (inconsistent equations).
If the lines are the same, there are infinitely many solutions (dependent equations).

Solving Graphically

Forming Simultaneous Equations from Context

Introduce letters (e.g. x, y) to represent the unknowns, with clear units.
Write two equations based on the given information (e.g. costs, totals, ratios).
Solve the equations simultaneously, then answer the question in context (e.g. cost of an apple is 40p).
Sometimes you need to find another quantity (e.g. product xy) after solving.

Quadratic Simultaneous Equations

Involve one linear equation and one quadratic (or non-linear) equation (e.g. x² + y² = 25 and y - 2x = 5).
Solve by substitution: rearrange the linear equation to y = ... (or x = ...) and substitute into the quadratic.
Expand and solve the resulting quadratic equation (may factorise, use formula, or complete square).
Substitute each x-value back into the linear equation to find the corresponding y-value.
Present solutions as pairs (e.g. x = 0, y = 5 or x = -4, y = -3).
If the quadratic has a repeated root, the line is tangent to the curve (one intersection).
If the quadratic has no real roots, the line and curve do not intersect (no solutions).

Line meets curve

Special Cases in Quadratic Simultaneous Equations

If the linear equation is not in the form y = ... or x = ..., rearrange it first before substituting.
For equations like xy = 3 and x + y = 4, rearrange either to y = 4 - x or y = 3/x and substitute.
Be careful with fractions: multiply through to clear denominators when necessary.
Always check solutions satisfy both original equations.

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练习题

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1.For linear simultaneous equations, which method involves making the coefficients of one variable the same in both equations and then adding or subtracting to eliminate that variable?
Easy
- AElimination method
- BSubstitution method
- CGraphical method
- DTrial and error method
2.When solving linear simultaneous equations graphically, the solution is given by the coordinates of the point where the lines:
Easy
- AIntersect
- BAre parallel
- CAre coincident
- DCross the x-axis
3.To solve the simultaneous equations 3x + 2y = 11 and 2x - y = 5 by elimination, you could multiply the second equation by 2 and then:
Medium
- AAdd to the first equation
- BSubtract from the first equation
- CMultiply by the first equation
- DDivide by the first equation
4.Solve the simultaneous equations: 2x + y = 7, 3x - y = 8. What is the value of x?
Medium
- A3
- B2
- C1
- D4
5.Solve the simultaneous equations: 3x - 8y = 22, x + 4y = 4. What is the value of y?
Medium
- A-0.5
- B-1
- C1
- D2
6.3 apples and 5 bananas cost £1.80. 5 apples and 1 banana cost £2.30. If a is the price of an apple in pence and b is the price of a banana in pence, which pair of equations represents this situation?
Medium
- A3a + 5b = 180, 5a + b = 230
- B3a + 5b = 1.80, 5a + b = 2.30
- C3a + 5b = 180, 5a + b = 2.30
- D3a + 5b = 1.80, 5a + b = 230
7.Solve the simultaneous equations: 5x + 8y = 4, (1/2)x + 3y = 7. What is the value of x?
Medium
- A-4
- B4
- C20
- D-20
8.Solve algebraically the simultaneous equations: x² + y² = 25, y - 2x = 5. Which of the following is a solution?
Medium
- Ax = 0, y = 5
- Bx = 0, y = -5
- Cx = 4, y = 3
- Dx = -4, y = 3

更多题目

9.The straight line y = 3x + 5 intersects the curve y = 2x² - 2x - 7 at points A and B. Find the x-coordinate of A (the smaller x).
Hard
- A-1.5
- B4
- C-2
- D3
10.The prices of two phones are in the ratio x:y. When the prices are both increased by £20, the ratio becomes 5:2. When the prices are both reduced by £5, the ratio becomes 5:1. Express the ratio x:y in its lowest terms.
Hard
- A4:1
- B3:1
- C5:2
- D2:1
11.Solve the simultaneous equations: y = 2x² - 7x + 4, y = 4x - 1. Which of the following is a pair of solutions?
Medium
- Ax = 0.5, y = 1 and x = 5, y = 19
- Bx = 1, y = 3 and x = 5, y = 19
- Cx = 0.5, y = 1 and x = 4, y = 15
- Dx = 1, y = 3 and x = 4, y = 15
12.Solve the simultaneous equations: y = 4 - x, x² + 2y² = 67. Which of the following is a solution?
Medium
- Ax = 7, y = -3
- Bx = 5, y = -1
- Cx = -7, y = 11
- Dx = 3, y = 1
13.Abdallah buys 3 cucumbers and 5 tomatoes for £14.50. Emerson buys 5 cucumbers and 4 tomatoes for £15.50. What is the cost of one cucumber in pounds?
Medium
- A1.50
- B2.00
- C1.00
- D2.50
14.Solve the simultaneous equations: x² + y² = 9, x + y = 2. Give your answers correct to 2 decimal places. Which of the following is a solution?
Hard
- Ax = 2.87, y = -0.87
- Bx = 2.87, y = 0.87
- Cx = -2.87, y = 4.87
- Dx = -0.87, y = 2.87
15.A rectangle has a perimeter of 30 cm. Its length is 3 cm more than its width. Let x be the width and y be the length. Which pair of equations represents this situation?
Easy
- A2x + 2y = 30, y = x + 3
- Bx + y = 30, y = x + 3
- C2x + 2y = 30, x = y + 3
- Dx + y = 30, x = y + 3
16.The diagram shows a triangle with base x cm and height (h - 3) cm, and a rectangle with base x cm and height h cm. The area of the triangle is 15 cm² and the area of the rectangle is 42 cm². What is the value of h?
Medium
- A10.5
- B7
- C10
- D8
17.Solve the simultaneous equations: 6x + 5y = 27, 5x - 3y = 44. What is the value of x?
Medium
- A7
- B3
- C5
- D9
18.Solve the simultaneous equations: 7x - 2y = 34, 3x + 5y = -3. What is the value of y?
Medium
- A-3
- B3
- C-5
- D5
19.Solve the simultaneous equations: 5a + 2c = 10, 2a - 4c = 7. What is the value of a?
Medium
- A2.25
- B2
- C3
- D1
20.Solve the simultaneous equations: x = 7 - 3y, x² - y² = 39. Which of the following is a solution?
Hard
- Ax = 7, y = 0 and x = -5, y = 4
- Bx = 6.25, y = 0.25 and x = -8, y = 5
- Cx = 7, y = 0 and x = 5, y = -4
- Dx = 6.25, y = 0.25 and x = 8, y = -5
21.Solve the simultaneous equations: 2x + 3y = -12, 5x + 2y = 14. What is the value of x?
Medium
- A6
- B-6
- C4
- D-4
22.The straight line y = 3x - 1 intersects the curve y = x² + x - 5 at points P and Q. Find the x-coordinate of P (the smaller x). Give your answer correct to 2 decimal places.
Hard
- A-1.24
- B-2.24
- C1.24
- D2.24
23.There are r red counters and g green counters in a bag. The probability of picking a green counter is 3/7. After adding 2 red and 3 green counters, the probability of green becomes 6/13. Find the original number of red counters.
Hard
- A9
- B12
- C6
- D3
24.Solve the simultaneous equations: x² + y² = 36, x = 2y + 6. Which of the following is a solution?
Hard
- Ax = 6, y = 0
- Bx = 0, y = -3
- Cx = 6, y = 0 and x = -3.6, y = -4.8
- Dx = 6, y = 0 and x = 3.6, y = 4.8
25.Solve the simultaneous equations: 1/2 x - 3y = 9, 5x + y = 28. What is the value of x?
Medium
- A6
- B8
- C10
- D12
26.Solve the simultaneous equations: 2x + 7y = 17, 5x + 3y = -1. What is the value of y?
Medium
- A3
- B-3
- C2
- D-2
27.What is the first step in solving linear simultaneous equations by elimination?
Easy
- AMake the coefficients of one variable the same in both equations
- BAdd the two equations together
- CSubstitute one equation into the other
- DPlot both equations on a graph
28.Which of the following is a correct method to eliminate y from these equations? 3x + 2y = 11 and 2x - y = 5
Medium
- AMultiply the second equation by 2 and add to the first
- BMultiply the second equation by 2 and subtract from the first
- CMultiply the first equation by 2 and add to the second
- DMultiply the first equation by 2 and subtract from the second
29.The diagram shows a triangle with base x cm and height (h - 3) cm, and a rectangle with base x cm and height h cm. The area of the triangle is 15 cm² and the area of the rectangle is 42 cm². Which equation gives the area of the triangle?
Easy
- A1/2×x×(h - 3) = 15
- Bx×(h - 3) = 15
- C1/2×x×h = 15
- Dx×h = 15
30.The prices of two phones are in the ratio x:y. When both prices are increased by £20, the ratio becomes 5:2. When both prices are reduced by £5, the ratio becomes 5:1. Find the value of x.
Hard
- A80
- B20
- C100
- D40
31.Solve the simultaneous equations: 3x + 2y = 11 and 2x - y = 5. What is the value of y?
Medium
- A1
- B2
- C3
- D-1
32.Solve the simultaneous equations: 5x + 2y = 11 and 4x - 3y = 18. What is the value of x?
Medium
- A3
- B-3
- C2
- D-2
33.Solve the simultaneous equations: y = 5x² + 4x - 19 and y = 4x + 1. Which of the following is a solution?
Hard
- Ax = 2, y = 9
- Bx = -2, y = -7
- Cx = 2, y = -7
- Dx = -2, y = 9
34.Solve the simultaneous equations: x + y = 7 and x - y = 3. What is the value of x?
Easy
- A5
- B2
- C4
- D3
35.When solving simultaneous equations graphically, the solution is given by the coordinates of the point where the lines cross.
Easy
True or false?
36.3 apples and 5 bananas cost £1.80. 5 apples and 1 banana cost £2.30. If a is the price of an apple in pence and b is the price of a banana in pence, which pair of equations represents this situation? (Select all that apply)
Medium
- A3a + 5b = 180
- B3a + 5b = 1.80
- C5a + b = 230
- D5a + b = 2.30
- Ea + b = 180
37.The prices of two phones are in the ratio x:y. When the prices are both increased by £20, the ratio becomes 5:2. When the prices are both reduced by £5, the ratio becomes 5:1. Which pair of equations correctly models this situation?
Medium
- A(x+20)/(y+20) = 5/2 and (x-5)/(y-5) = 5/1
- B(x+20)/(y+20) = 2/5 and (x-5)/(y-5) = 1/5
- Cx/y = 5/2 and x/y = 5/1
- D(x+20)/(y-20) = 5/2 and (x-5)/(y+5) = 5/1
38.Solve the simultaneous equations: y = 2x² - 7x + 4, y = 4x - 1. Which of the following is a pair of solutions? (Select all that apply)
Hard
- Ax = 0.5, y = 1
- Bx = 2.5, y = 9
- Cx = 1, y = 3
- Dx = 5, y = 19
39.Match each step in solving the simultaneous equations 3x + 2y = 11 and 2x - y = 5 by elimination to its correct description.
Medium
- Multiply second equation by 2
- Add the two equations
- Solve 7x = 21
- Substitute x = 3 into an original equation
- 4x - 2y = 10
- 7x = 21
- x = 3
- y = 1
40.Arrange the steps for solving the simultaneous equations y = 2x² - 3x + 1 and y = 4x - 3 in the correct order.
Medium
- Set 2x² - 3x + 1 = 4x - 3
- Rearrange to 2x² - 7x + 4 = 0
- Factorise to (2x - 1)(x - 4) = 0
- Solve for x: x = 0.5 and x = 4
- Substitute into y = 4x - 3 to find y = -1 and y = 13
41.Which of the following are correct first steps when solving the simultaneous equations x² + y² = 25 and y - 2x = 5? (Select all that apply)
Medium
- ARearrange the linear equation to y = 2x + 5
- BSubstitute y = 2x + 5 into the quadratic equation
- CMultiply the linear equation by 2
- DSubtract the linear equation from the quadratic
42.When solving linear simultaneous equations by elimination, if the coefficients of the variable to be eliminated have the same sign, you should add the equations.
Easy
True or false?
43.Solve the simultaneous equations: y = 4 - x, x² + 2y² = 67. Give the positive y value, correct to 3 significant figures.
Medium
44.Solve the simultaneous equations: 4x + 3y = 10, 2x - 5y = 18. What is the value of y?
Medium
45.Complete the steps to solve the simultaneous equations y = 3x - 2 and y = x² - 2x + 2.
Medium
Set 3x - 2 = x² - 2x + 2. Rearrange to 0 = x² - 5x + 4. Factorise to (x - 1)(x - 4) = 0. So x = ____ or x = ____. Then y = ____ or y = ____.
46.A rectangle has a perimeter of 30 cm. Its length is 3 cm more than its width. Form a pair of simultaneous equations and solve them to find the length and width. Give your answer as 'length = ... cm, width = ... cm'.
Medium
47.Solve the simultaneous equations: 4x + 3y = 10, 2x - 5y = 18. What is the value of y?
Medium
- A-2
- B-1
- C0
- D-3

复习笔记

## Simultaneous Equations - Revision Notes ### Linear Simultaneous Equations - **What they are:** Two equations with two unknowns (x, y) that are both linear (no x², y², xy). - **Solving by Elimination:** 1. Make the coefficients of one variable the same in both equations (multiply if needed). 2. If signs are the same, **subtract** equations; if different, **add** equations to eliminate that variable. 3. Solve the resulting equation for the remaining variable. 4. Substitute back into one original equation to find the other variable. 5. Check both solutions satisfy the original equations. - **Solving by Substitution:** 1. Rearrange one equation to y = ... or x = ... 2. Substitute this expression into the other equation. 3. Solve for the variable, then substitute back to find the other. - **Solving Graphically:** Plot both lines; the intersection point gives the solution (x, y). - **Forming Equations:** Define variables (e.g., x = price of apple, y = price of banana) and write two equations from the problem context. Solve and answer in context with units. ### Quadratic Simultaneous Equations - **What they are:** One linear and one quadratic (e.g., x² + y² = 25) equation. - **Solving by Substitution (algebraic method):** 1. Rearrange the linear equation to y = ... or x = ... 2. Substitute into the quadratic equation. 3. Expand and solve the resulting quadratic (factorise or use formula). 4. Substitute each x (or y) back into the linear equation to find corresponding y (or x). 5. Present solutions as pairs (e.g., x = 0, y = 5 or x = -4, y = -3). - **Graphical Interpretation:** The intersection(s) of the line and curve give the solutions. A repeated root means the line is tangent; no real roots means no intersection. ### Key Points - Always check solutions satisfy **both** original equations. - Write final answers as clear pairs (x = ..., y = ...). - For word problems, answer the question asked (e.g., total cost, not just x and y).

Simultaneous Equations: Two or more equations that share variables; the solution satisfies all equations at once.
Elimination Method: A method to solve simultaneous equations by adding or subtracting the equations to remove one variable.
Substitution Method: A method to solve simultaneous equations by rearranging one equation and substituting it into the other.
Quadratic Simultaneous Equations: A pair of equations where at least one is quadratic (contains x², y², or xy).
Coefficient: The number multiplying a variable (e.g., in 3x, the coefficient is 3).
Point of Intersection: The point where two graphs meet; its coordinates are the solution to the simultaneous equations.

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