Simultaneous Equations | 联立方程

📚 Simultaneous Equations | 联立方程

Simultaneous equations are a set of equations containing multiple unknown variables, where all equations must be satisfied at the same time. In the Edexcel IGCSE Mathematics syllabus, solving these systems is a core skill, appearing in both non-calculator and calculator papers. This guide will walk you through the essential methods, typical pitfalls, and effective strategies to secure full marks.

联立方程是含有多个未知数的方程组，所有方程需要在同一时刻成立。在 Edexcel 爱德思 IGCSE 数学考纲中，解这类方程组是核心技能，在非计算器试卷和计算器试卷中都会出现。本文将带你梳理主要解法、常见错误以及获得满分的有效策略。

1. What Are Simultaneous Equations? | 什么是联立方程？

A linear simultaneous equation system in two unknowns typically looks like:

2x + 3y = 7
5x − 2y = 8

Here, x and y represent the same values in both equations. Solving the system means finding the exact pair (x, y) that makes both equations true. In the Edexcel syllabus, you must be able to solve such systems using algebraic methods, and also interpret them graphically.

一个含有两个未知数的线性联立方程组通常形如：

2x + 3y = 7
5x − 2y = 8

这里的 x 和 y 在两个方程中代表相同的数值。解方程组就是找到同时满足两个方程的精确数值对 (x, y)。在爱德思大纲中，你需要掌握用代数方法求解，并能从图像上理解方程组的含义。

2. Solving by Substitution | 代入消元法

The substitution method works best when one variable has a coefficient of 1. Rearrange one equation to make either x or y the subject, then substitute this expression into the other equation. This produces a single linear equation in one unknown.

代入消元法适合某个变量系数为 1 的情况。先从一个方程中表示出 x 或 y，再将这个表达式代入另一个方程，从而得到一个一元一次方程。

Example | 例题：
Solve the system:

y = 2x − 1
x + 3y = 11

Substitute y = 2x − 1 into the second equation:

x + 3(2x − 1) = 11

Expand and simplify:

x + 6x − 3 = 11 → 7x = 14 → x = 2

Now find y: y = 2(2) − 1 = 3. Solution: (2, 3).

将 y = 2x − 1 代入第二个方程：

x + 3(2x − 1) = 11

展开并化简：

x + 6x − 3 = 11 → 7x = 14 → x = 2

再求 y：y = 2(2) − 1 = 3。解为 (2, 3)。

3. Solving by Elimination | 加减消元法

Elimination is often the fastest for both linear equations in standard form. Multiply one or both equations so that the coefficients for either x or y are equal in magnitude, then add or subtract to eliminate that variable. After solving for one variable, substitute back to find the other.

当两个方程都是标准形式时，加减消元法通常最快。将其中一个或两个方程乘以适当的数，使 x 或 y 的系数绝对值相等，然后相加或相减以消去该变量。求出其中一个未知数后，代回原方程求另一个。

Example | 例题：
Solve:

3x + 2y = 12
2x − y = 3

Multiply the second equation by 2 to equalise y coefficients:

4x − 2y = 6

Now add the equations (since +2y and −2y cancel):

3x + 2y + 4x − 2y = 12 + 6 → 7x = 18 → x = 18⁄7

Substitute into 2x − y = 3:

2(18⁄7) − y = 3 → 36⁄7 − y = 3 → y = 36⁄7 − 21⁄7 = 15⁄7

Solution: (18⁄7, 15⁄7).

将第二个方程乘以 2，使 y 的系数绝对值相等：

4x − 2y = 6

两式相加（因为 +2y 和 −2y 相消）：

3x + 2y + 4x − 2y = 12 + 6 → 7x = 18 → x = 18⁄7

代入 2x − y = 3：

2(18⁄7) − y = 3 → 36⁄7 − y = 3 → y = 36⁄7 − 21⁄7 = 15⁄7

解为 (18⁄7, 15⁄7)。

4. Graphical Method | 图像法

When solving graphically, plot both straight lines on the same coordinate grid. The intersection point gives the simultaneous solution. For exact answers you may need to read the coordinates carefully, or use a calculator if allowed. However, algebraic methods are generally preferred for non-integer answers.

用图像法求解时，在同一坐标网格中画出两条直线。两条线的交点就是联立方程的解。对于精确答案，你可能需要仔细读取坐标，或在允许时使用计算器。不过对于非整数解，代数方法通常更优。

Example | 例题：
Draw the lines y = 2x − 1 and x + 3y = 11 (rearranged as y = (11 − x)⁄3). Their intersection is (2, 3), matching the substitution example above.

画出直线 y = 2x − 1 和 x + 3y = 11（改写为 y = (11 − x)⁄3）。它们的交点为 (2, 3)，与前面代入法例题一致。

Remember the three cases: if lines meet at one point, there is a unique solution; if lines are parallel, there are no solutions; if lines coincide, there are infinitely many solutions.

记住三种情况：如果两条直线相交于一点，则方程组有唯一解；如果平行，则无解；如果重合，则有无穷多解。

5. Linear and Quadratic Systems | 线性与二次方程组

Edexcel IGCSE often asks you to solve a linear equation together with a quadratic. These systems produce up to two solutions. The method is always substitution: make a variable the subject from the linear equation, then substitute into the quadratic. This yields a quadratic in one variable, which you solve by factorising or using the quadratic formula.

爱德思 IGCSE 时常要求解一个线性方程与一个二次方程组成的方程组。这类系统最多有两个解。解法总是代入法：先从线性方程中表示出某个变量，再代入二次方程，得到一个一元二次方程，通过因式分解或求根公式求解。

Example | 例题：
Solve:

y = 2x + 1
x² + y² = 10

Substitute y = 2x + 1 into x² + y² = 10:

x² + (2x + 1)² = 10

Expand: x² + 4x² + 4x + 1 = 10 → 5x² + 4x − 9 = 0

Factorise: (5x + 9)(x − 1) = 0 → x = −9⁄5 or x = 1

Find y for each: for x = 1, y = 3; for x = −9⁄5, y = 2(−9⁄5) + 1 = −18⁄5 + 5⁄5 = −13⁄5. Solutions: (1, 3) and (−9⁄5, −13⁄5).

将 y = 2x + 1 代入 x² + y² = 10：

x² + (2x + 1)² = 10

展开：x² + 4x² + 4x + 1 = 10 → 5x² + 4x − 9 = 0

因式分解：(5x + 9)(x − 1) = 0 → x = −9⁄5 或 x = 1

分别求 y：当 x = 1 时，y = 3；当 x = −9⁄5 时，y = 2(−9⁄5) + 1 = −18⁄5 + 5⁄5 = −13⁄5。解为 (1, 3) 和 (−9⁄5, −13⁄5)。

6. Setting Up From Word Problems | 从文字题建立方程

In exam problems, you often need to translate information into a pair of equations. Define two unknown variables, then carefully express each condition as an equation. Common contexts include money, ages, mixtures, and geometric shapes.

考试中常常需要将文字信息转化为两个方程。先定义两个未知量，然后仔细将每个条件写成方程。常见情境包括金钱、年龄、混合物和几何图形。

Worked Example | 例题：
Two pens and three notebooks cost £7. Four pens and one notebook cost £9. Find the cost of one pen (p) and one notebook (n).

两支笔和三本笔记本共 7 英镑。四支笔和一本笔记本共 9 英镑。求一支笔 (p) 和一本笔记本 (n) 的价格。

Equations:

2p + 3n = 7
4p + n = 9

Using elimination: multiply the first by 2: 4p + 6n = 14. Subtract the second: (4p + 6n) − (4p + n) = 14 − 9 → 5n = 5 → n = 1. Substitute into 4p + 1 = 9 → 4p = 8 → p = 2. So a pen costs £2 and a notebook £1.

方程组：

2p + 3n = 7
4p + n = 9

用消元法：第一个方程乘以 2：4p + 6n = 14。减去第二个方程：(4p + 6n) − (4p + n) = 14 − 9 → 5n = 5 → n = 1。代入 4p + 1 = 9 → 4p = 8 → p = 2。所以一支笔 2 英镑，一本笔记本 1 英镑。

7. Check Your Solution | 验证解的正确性

Always substitute the solution into both original equations. If both sides balance, your answer is correct. For linear systems, a quick check in both equations takes only seconds and prevents careless errors.

务必把解代入原方程验证。如果两边数值相等，说明答案正确。对于线性系统，在两个方程中快速验证只需要几秒钟，能避免粗心造成的错误。

For the previous pen/notebook example, check: 2(2) + 3(1) = 4 + 3 = 7 ✓; 4(2) + 1 = 8 + 1 = 9 ✓.

对于前一个笔/笔记本例题，验证：2(2) + 3(1) = 4 + 3 = 7 ✓；4(2) + 1 = 8 + 1 = 9 ✓。

8. Edexcel Exam Tips | 爱德思考场提示

Read the question carefully – decide which method (substitution or elimination) is fastest. If a variable is already on its own, substitution is natural.

仔细读题 – 判断哪种方法（代入还是消元）最快。如果某个变量已经单独出现在等式一侧，自然用代入法。
Show all working – method marks are awarded even if the final answer is wrong. Write each step clearly.

写出完整过程 – 即使答案错误，方法分也会给到。每一步都要写清楚。
Watch out for signs – subtracting equations can cause sign errors. Use brackets and align like terms.

注意符号 – 两式相减容易出错。使用括号并对齐同类项。
For quadratic systems – expect two solutions, but note that occasionally the quadratic has a repeated root, yielding only one point.

对于二次系统 – 预期有两个解，但偶尔二次方程有重根，只会得到一个点。
Use a calculator for arithmetic – but only after you have set up the correct algebra. Do not rely on the calculator to solve the equation unless you are using a CAS (not allowed in IGCSE).

用计算器做算术 – 但只有在列出正确的代数式之后。不要依赖计算器解方程，除非是在 IGCSE 中不允许的 CAS 计算器。
Graphical answers – when asked to draw graphs, use a sharp pencil and ruler. Give coordinates as exact fractions or decimals if required.

图像法答案 – 需要画图时，用削尖的铅笔和直尺。若要求精确，请以分数或小数形式给出坐标。

9. Common Mistakes | 常见错误

Incorrect substitution – forgetting to multiply every term when substituting a negative expression.

代入错误 – 代入一个带负号的表达式时，忘记乘以每一个项。
Choosing the wrong variable – wasting time making a variable the subject when elimination is easier.

选错变量 – 在消元法更容易时，却浪费时间表示某个变量。
Arithmetic errors – basic addition, subtraction, or multiplication mistakes. Double-check each step.

计算错误 – 基本的加减乘除失误。每一步都要复核。
No solution check – skipping the verification stage can lose easy marks.

不检查 – 跳过验证步骤会丢失容易拿到的分。
Forgetting units in word problems – always include the correct unit (₹, £, cm, m², etc.) in your final answer.

文字题忘记单位 – 最终答案一定要带上正确单位（₹、£、cm、m² 等）。

10. Practice Problem | 练习题目

Solve the following system:

5x − 2y = 4
3x + 7y = 13

Solution | 解答：
Use elimination. Multiply the first equation by 7 and the second by 2 to equalise y coefficients (both 14y, but opposite signs? Let’s see: first ×7: 35x − 14y = 28; second ×2: 6x + 14y = 26. Now add: 35x − 14y + 6x + 14y = 28 + 26 → 41x = 54 → x = 54⁄41.

Substitute into first: 5(54⁄41) − 2y = 4 → 270⁄41 − 2y = 4 → −2y = 4 − 270⁄41 = 164⁄41 − 270⁄41 = −106⁄41 → y = 53⁄41.

Solution: (54⁄41, 53⁄41). Verify: 3(54⁄41) + 7(53⁄41) = 162⁄41 + 371⁄41 = 533⁄41 = 13 ✓.

用消元法解：第一个方程乘以 7，第二个方程乘以 2，使 y 系数绝对值相等（都是 14y，但符号相反）：第一个×7：35x − 14y = 28；第二个×2：6x + 14y = 26。相加：35x − 14y + 6x + 14y = 28 + 26 → 41x = 54 → x = 54⁄41。

代入第一个方程：5(54⁄41) − 2y = 4 → 270⁄41 − 2y = 4 → −2y = 4 − 270⁄41 = 164⁄41 − 270⁄41 = −106⁄41 → y = 53⁄41。

解为 (54⁄41, 53⁄41)。验证：3(54⁄41) + 7(53⁄41) = 162⁄41 + 371⁄41 = 533⁄41 = 13 ✓。

11. Advanced: Simultaneous Equations in Three Variables | 进阶：三元联立方程

For Higher Tier IGCSE, you may occasionally meet three equations in three variables. The strategy extends elimination step by step: reduce the system from 3 to 2 variables, then to 1. For example, eliminate z from two pairs of equations, leaving two equations in x and y, which you solve as usual.

在 IGCSE 高级个别科目（Higher Tier）中，偶尔会遇到含有三个变量的三个方程。策略是逐步消元：先把三元方程组化为二元，再化为一元。例如，从两组方程中消去 z，留下关于 x 和 y 的两个方程，再按常规方法求解。

Quick Example | 简短例题：

x + y + z = 6
x − y + 2z = 5
2x + y − z = 3

Subtract the first from the second: (x − y + 2z) − (x + y + z) = 5 − 6 → −2y + z = −1 → z = 2y − 1. Also subtract 2×(first) from the third: (2x + y − z) − 2(x + y + z) = 3 − 12 → −y −3z = −9 → y + 3z = 9. Substitute z = 2y − 1 into y + 3(2y − 1) = 9 → y + 6y − 3 = 9 → 7y = 12 → y = 12⁄7. Then z = 2(12⁄7) − 1 = 24⁄7 − 7⁄7 = 17⁄7. Finally x = 6 − y − z = 6 − 12⁄7 − 17⁄7 = 42⁄7 − 29⁄7 = 13⁄7. Solution: (13⁄7, 12⁄7, 17⁄7). Check in second: 13⁄7 − 12⁄7 + 2(17⁄7) = 13⁄7 − 12⁄7 + 34⁄7 = 35⁄7 = 5 ✓.

将第一个方程从第二个方程中减去：(x − y + 2z) − (x + y + z) = 5 − 6 → −2y + z = −1 → z = 2y − 1。再将 2×(第一个) 从第三个方程中减去：(2x + y − z) − 2(x + y + z) = 3 − 12 → −y −3z = −9 → y + 3z = 9。将 z = 2y − 1 代入 y + 3(2y − 1) = 9 → y + 6y − 3 = 9 → 7y = 12 → y = 12⁄7。则 z = 2(12⁄7) − 1 = 24⁄7 − 7⁄7 = 17⁄7。最后 x = 6 − y − z = 6 − 12⁄7 − 17⁄7 = 42⁄7 − 29⁄7 = 13⁄7。解为 (13⁄7, 12⁄7, 17⁄7)。验证第二个方程：13⁄7 − 12⁄7 + 2(17⁄7) = 13⁄7 − 12⁄7 + 34⁄7 = 35⁄7 = 5 ✓。

12. Summary | 总结

Mastering simultaneous equations gives you a powerful problem‑solving tool. In the Edexcel IGCSE exam, systems appear in algebra sections, in word problems, and even inside geometry or coordinate geometry questions. Practice both substitution and elimination until each becomes automatic. For linear‑quadratic systems, always substitute and solve the resulting quadratic. Finally, never forget to check your answers in the original equations.

掌握联立方程能为你提供一个强大的解题工具。在 Edexcel IGCSE 考试中，方程组出现在代数部分、文字题、甚至几何或坐标几何题目中。反复练习代入法和消元法，直至两种方法都变得自动熟练。对于线性‑二次系统，始终用代入法并解所得二次方程。最后，千万不要忘记在原始方程中验证答案。

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Simultaneous Equations | 联立方程