Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are one of the most important building blocks in KS3 mathematics. In the Cambridge curriculum, students are expected to solve linear equations with confidence, using a logical, step-by-step method. This revision article covers the core techniques you need for classwork, homework and examinations.

一元一次方程是 KS3 数学中最重要的基础模块之一。在剑桥课程中，学生需要能够自信地、按逻辑、分步骤地解一元一次方程。本文涵盖课堂学习、作业和考试所需的核心技巧。

1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an equation in which the unknown variable, usually written as x, appears only to the first power. There are no x² terms, no square roots of x, and no products of x with itself. The word 'linear' tells you that the graph of the equation would be a straight line.

一元一次方程是未知数（通常写作 x）只出现一次幂的方程。没有 x² 项，没有 x 的平方根，也没有 x 与自身的乘积。“线性”一词说明该方程的图像是一条直线。

ys to find the value of x that makes the equation true. This value is called the solution.

在剑桥 KS3 阶段，你会遇到像 x + 3 = 7、2x = 10、5x − 4 = 11 和 3(x + 2) = 15 这样的方程。任务始终是求出使等式成立的 x 值，这个值叫做方程的解。

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2. The Balance Method | 天平法

A good way to understand solving equations is to imagine a balance scale. The left side and the right side of the equation must stay equal, so whatever operation you do to one side, you must also do to the other side. This rule is the foundation of all linear equation work.

理解解方程的一个好方法是想象一架天平。方程左边和右边必须保持相等，所以对一边进行任何运算，对另一边也必须进行相同运算。这条规则是所有一元一次方程运算的基础。

For example, if you subtract 3 from the left side, you must subtract 3 from the right side as well. If you multiply one side by 5, you must multiply the other side by 5. This keeps the equation balanced while you isolate the unknown.

例如，如果你从左边减去 3，就必须从右边也减去 3。如果你把一边乘以 5，就必须把另一边也乘以 5。这样可以在分离未知数的同时保持方程平衡。

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3. Solving with Addition and Subtraction | 用加法与减法解方程

ns, so they undo each other. For example, to solve x + 5 = 12, subtract 5 from both sides.

当一个数加到 x 上或从 x 中减去时，使用逆运算来消除它。加法和减法互为逆运算，所以它们可以相互抵消。例如，解 x + 5 = 12，两边同时减去 5。

x + 5 − 5 = 12 − 5

can check by substituting 7 back into the original equation: 7 + 5 = 12, which is correct.

化简得到 x = 7。你可以把 7 代回原方程检验：7 + 5 = 12，这是正确的。

to get x = 13. Always use the opposite operation to move a number across the equals sign.

对于 x − 4 = 9 这样的方程，两边同时加 4 得到 x = 13。始终使用相反的运算把数字移到等号的另一边。

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4. Solving with Multiplication and Division | 用乘法与除法解方程

If x is multiplied by a number, divide both sides by that number. If x is divided by a number, multiply both sides by that number. Multiplication and division are also inverse operations.

如果 x 乘以一个数，两边同时除以这个数。如果 x 除以一个数，两边同时乘以这个数。乘法和除法同样互为逆运算。

Example: solve 4x = 28. To isolate x, divide both sides by 4.

例如：解 4x = 28。为了分离 x，两边同时除以 4。

4x ÷ 4 = 28 ÷ 4

So x = 7. This means 4 × 7 = 28, which confirms the solution.

所以 x = 7。这意味着 4 × 7 = 28，验证了解的正确性。

Example: solve x ÷ 5 = 3. Multiply both sides by 5.

例如：解 x ÷ 5 = 3。两边同时乘以 5。

x ÷ 5 × 5 = 3 × 5

So x = 15. In this case, 15 divided by 5 equals 3, which matches the original equation.

所以 x = 15。在这种情况下，15 除以 5 等于 3，与原方程一致。

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5. Two-Step Equations | 两步方程

Many KS3 equations need two steps. Always undo addition or subtraction before multiplication or division. This order is important because it reverses the normal order of operations.

许多 KS3 方程需要两步。总是先消去加法或减法，再处理乘法或除法。这个顺序很重要，因为它与正常运算顺序相反。

Example: solve 2x + 3 = 11. Step 1 is to subtract 3

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