Solving Linear Equations with Unknowns on Both Sides | 解等号两边含未知数的一元一次方程

📚 Solving Linear Equations with Unknowns on Both Sides | 解等号两边含未知数的一元一次方程

2x + 9. You will learn how to expand brackets, collect like terms, and use the balancing method to find the value of x with confidence.

在剑桥 KS3 数学课程中，解一元一次方程是代数部分的核心技能。本文重点讲解未知数出现在等号两边的方程，例如 5x + 3 = 2x + 9。你将学习如何先去括号、合并同类项，并运用平衡法自信地求出 x 的值。

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1. What Is a Linear Equation? | 什么是线性方程

y raised to the power 1. For example, 3x + 2 = 11 is linear, but x² + 2 = 6 is not linear.

线性方程是指未知数（通常为 x）的指数仅为 1 的方程。例如 3x + 2 = 11 是线性方程，而 x² + 2 = 6 不是线性方程。

In KS3, linear equations often include brackets, fractions, or unknowns on both sides of the equals sign. The goal is always the same: isolate x on one side.

在 KS3 阶段，线性方程通常包含括号、分数或等号两边都有未知数。目标始终相同：将 x 单独留在等式的一边。

Examples of linear equations are 4x - 7 = 13, 2(x + 5) = 3x - 2, and x/3 + 1 = 5.

线性方程的例子有 4x - 7 = 13、2(x + 5) = 3x - 2 和 x/3 + 1 = 5。

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2. The Balancing Method | 平衡法

An equation is like a balance scale. Whatever you do to one side, you must do to the other side to keep the equation balanced.

方程就像一架天平。你对等式一边做的任何运算，必须对另一边也做同样的运算，才能保持等式平衡。

3 = 2x + 9, first remove 2x from both sides: 5x - 2x + 3 = 2x - 2x + 9, giving 3x + 3 = 9.

解 5x + 3 = 2x + 9 时，先从两边减去 2x：5x - 2x + 3 = 2x - 2x + 9，得到 3x + 3 = 9。

Then subtract 3 from both sides to get 3x = 6, and divide both sides by 3 to find x = 2.

然后从两边减去 3，得到 3x = 6，再将两边同时除以 3，得到 x = 2。

This method works for every linear equation, no matter how complicated it looks at first.

无论方程起初看起来多么复杂，这种方法都适用于所有一元一次方程。

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3. Expanding Brackets First | 先去括号

ys expand them before collecting like terms. Use the distributive law: a(b + c) = ab + ac.

出现括号时，一定要先展开括号，再合并同类项。使用分配律：a(b + c) = ab + ac。

For example, solve 3(x + 2) = 2x + 10. Expand the left side to get 3x + 6 = 2x + 10.

例如，解方程 3(x + 2) = 2x + 10。展开左边得到 3x + 6 = 2x + 10。

Then subtract 2x from both sides: x + 6 = 10. Finally subtract 6 from both sides: x = 4.

然后从两边减去 2x：x + 6 = 10。最后从两边减去 6：x = 4。

If the brackets contain subtraction, such as 2(x - 4), expand to 2x - 8, not 2x - 4.

如果括号内是减法，例如 2(x - 4)，展开后应为 2x - 8，而不是 2x - 4。

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4. Collecting Like Terms | 合并同类项

Like terms have exactly the same letter and power. You can combine 5x and -2x, but you cannot combine 5x and 5.

同类项具有完全相同的字母和指数。你可以合并 5x 和 -2x，但不能合并 5x 和 5。

If an equation has x terms on both sides and number terms on both sides, group the x terms on one side and the number terms on the other side.

如果方程两边都有含 x 的项和数字项，就把含 x 的项移到一边，数字项移到另一边。

both sides to get 3x + 7 = -8. Then subtract 7 from both sides to get 3x = -15, so x = -5.

对于 4x + 7 = x - 8，从两边减去 x 得到 3x + 7 = -8。再从两边减去 7 得到 3x = -15，所以 x = -5。

Working carefully with positive and negative signs is essential when collecting like terms.

合并同类项时，仔细处理正负号是非常重要的。

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5. Unknowns on Both Sides | 未知数在等号两边

The most efficient method is to remove the smaller x term first. This keeps the coefficient of x positive, which reduces sign errors.

最高效的方法是先消去较小的 x 项。这样可以使 x 的系数保持为正，减少符号错误。

e 7x - 3 = 2x + 12. The smaller x term is 2x, so subtract 2x from both sides: 5x - 3 = 12.

解方程 7x - 3 = 2x + 12。较小的 x 项是 2x，所以从两边减去 2x：5x - 3 = 12。

Add 3 to both sides: 5x = 15. Divide both sides by 5: x = 3.

两边加 3：5x = 15。两边除以 5：x = 3。

If the smaller x term is on the left instead, still remove it from both sides. The process is unchanged.

如果较小的 x 项在左边，同样从两边减去它。解题过程完全不变。

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6. Dealing with Fractions | 处理分数

If an equation contains fractions, multiply every term on both sides by the lowest common denominator to clear the fractions.

如果方程含有分数，将两边每一项都乘以最小公分母，以消去分数。

For example, solve x/2 + 3 = x/4 + 5. The lowest common denominator of 2

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