Changing the Subject of a Formula | 变换公式的主项

📚 Changing the Subject of a Formula | 变换公式的主项

Changing the subject of a formula, also called rearranging a formula, is one of the most important algebraic techniques in IGCSE Mathematics. It allows you to take an existing relationship and rewrite it so that any chosen variable is expressed in terms of the others. This is used constantly in algebra, geometry, physics and finance.

变换公式的主项，也叫重排公式，是 IGCSE 数学中最重要的代数技巧之一。它让你能够依据已有的关系式，把其中任何一个选定的变量写成由其它变量表达的式子。这在代数、几何、物理和金融中都会被反复使用。

Many students find rearranging formulas confusing because the letters look abstract. However, the same rules apply as for solving ordinary equations: whatever you do to one side, you must do to the other side. Once you master the inverse operations and the correct order of steps, every formula becomes a simple puzzle.

许多学生觉得变换公式很困惑，因为字母看起来比较抽象。然而，它所遵循的规则和普通解方程完全相同：你对一边做的操作，也必须对另一边做同样的操作。一旦你掌握了逆运算和正确的步骤顺序，每个公式都会变成一个简单的拼图。

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1. The Subject of a Formula | 什么是公式的主项

The subject of a formula is the variable that is written alone on one side of the equals sign, usually on the left. In the formula C = 2πr, the subject is C because it is isolated. The right-hand side, 2πr, gives the expression that defines C.

公式的主项是指单独写在等号一侧的变量，通常在左侧。在公式 C = 2πr 中，主项是 C，因为它被单独隔离了出来。右侧的 2πr 是定义 C 的表达式。

To change the subject means to choose a different variable and rearrange the formula so that this new variable is alone on one side. For example, if we want r to be the subject of C = 2πr, we can rearrange to obtain r = C ÷ (2π).

变换主项意味着选择另一个变量，并重排公式，使这个新变量单独位于一侧。例如，如果我们想让 r 成为公式 C = 2πr 的主项，可以重排得到 r = C ÷ (2π)。

It is essential to write the final answer with the new subject clearly on the left and no remaining instances of that variable on the right. If the variable appears more than once, you must collect the terms together before isolating it.

写最终答案时，必须让新主项清晰地位于左侧，并且右侧不再有这个变量出现。如果变量出现多次，必须先归类合并所有含它的项，然后再进行隔离。

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2. Inverse Operations | 逆运算

Every operation has an inverse operation that undoes it. Addition and subtraction are inverses of each other; multiplication and division are inverses of each other; powers and roots are inverses of each other. To move a term across the equals sign, apply the inverse operation to both sides.

每个运算都有一个能抵消它的逆运算。加法和减法互为逆运算；乘法和除法互为逆运算；幂和根互为逆运算。要将某一项移到等号另一边，就对两边同时施加逆运算。

Operation / 运算	Inverse / 逆运算
+	−
−	+
×	÷
÷	×
square ²	square root √
square root √	square ²
cube ³	cube root ∛

The golden rule is balance. If you subtract 5 from the left side, you must subtract 5 from the right side too. If you divide the left side by 4, you must divide the right side by 4. Keeping the equation balanced guarantees that the new formula is equivalent to the original.

黄金法则就是保持平衡。如果你从左边减去 5，也必须从右边减去 5。如果你把左边除以 4，也必须把右边除以 4。保持等式平衡能保证新公式与原公式等价。

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3. One-Step Rearrangements | 一步重排

When the desired variable is involved in only a single addition, subtraction, multiplication or division, one inverse operation is enough. Start with the simplest examples.

当目标变量只涉及一次加法、减法、乘法或除法时，一步逆运算就足够了。我们先从最简单的例子开始。

Example 1: Make x the subject of y = x + 7.

例1：在 y = x + 7 中求 x。

y = x + 7
y − 7 = x
x = y − 7

Since 7 is added to x, we subtract 7 from both sides. The final answer is x = y − 7.

因为 x 加上了 7，所以两边同时减去 7。最终答案是 x = y − 7。

Example 2: Make p the subject of A = 5p.

例2：在 A = 5p 中求 p。

A = 5p
A ÷ 5 = p
p = A/5

Here p is multiplied by 5, so we divide both sides by 5. Note that the answer can also be written as A/5 or A ÷ 5.

这里 p 乘以了 5，所以两边同时除以 5。注意答案也可以写成 A/5 或 A ÷ 5。

Example 3: Make t the subject of D = t − 12.

例3：在 D = t − 12 中求 t。

D = t − 12
D + 12 = t
t = D + 12

Because 12 was subtracted from t, we add 12 to both sides.

因为 t 减去了 12，所以两边同时加上 12。

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4. Two-Step Rearrangements | 两步重排

Most formulas require at least two inverse operations. The order of these operations is crucial: first undo any addition or subtraction, then undo any multiplication or division. This is the reverse of the order of operations used when evaluating the original formula.

大多数公式至少需要两步逆运算。这些运算的顺序至关重要：先抵消加法和减法，再抵消乘法和除法。这与计算原公式时使用的运算顺序正好相反。

Example: Make x the subject of y = 3x + 2.

例：在 y = 3x + 2 中求 x。

y = 3x + 2
y − 2 = 3x
(y − 2) ÷ 3 = x
x = (y − 2)/3

In the original formula, we first multiply x by 3 and then add 2. To reverse this, we first subtract 2 and then divide by 3. The brackets around y − 2 are essential because the whole expression is divided by 3.

在原来的公式中，我们先让 x 乘以 3，然后再加上 2。要逆转这个过程，我们应先减去 2，再除以 3。括号 y − 2 必不可少，因为整个表达式都要除以 3。

Example: Make m the subject of E = (mgh)/2.

例：在 E = (mgh)/2 中求 m。

E = (mgh)/2
2E = mgh
2E ÷ (gh) = m
m = 2E/(gh)

Here m is multiplied by g and h, then the whole product is divided by 2. We multiply both sides by 2 first, then divide by gh.

这里 m 先乘以 g 和 h，然后整个乘积再除以 2。我们先把两边乘以 2，再除以 gh。

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5. Subject in a Denominator | 主项在分母中

Sometimes the variable you want is in the denominator of a fraction. The first step is to multiply both sides by the entire denominator. This moves the variable out of the denominator.

有时你想要的变量出现在一个分数的分母中。第一步是把两边都乘以整个分母。这样就能把变量从分母中移出来。

Example: Make x the subject of y = a/x.

例：在 y = a/x 中求 x。

y = a/x
y × x = a
yx = a
x = a/y

Multiplying both sides by x cancels the denominator. Then dividing both sides by y gives x alone.

两边同乘 x 可以约去分母。然后再两边同除以 y，就能单独得到 x。

Example: Make t the subject of V = k/(t + 3).

例：在 V = k/(t + 3) 中求 t。

V = k/(t + 3)
V(t + 3) = k
t + 3 = k/V
t = k/V − 3

When multiplying by t + 3, remember to multiply the whole left-hand side and the whole right-hand side. Then divide by V and finally subtract 3.

乘以 t + 3 时，要记住整个左边和整个右边都要相乘。然后再除以 V，最后减去 3。

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6. Subject Appears Twice | 主项出现两次

If the variable you want appears in two or more terms, you cannot isolate it with inverse operations alone. Instead, gather every term containing that variable on one side of the equation and factorise it out. This technique is common when the subject appears in the numerator and denominator of a fraction.

如果你想要的变量出现在两个或更多项中，你不能仅靠逆运算来隔离它。相反，要把所有含该变量的项集中到等式的一边，然后提取公因式。当主项同时出现在分数的分子和分母中时，这种技巧非常常见。

Example: Make x the subject of y = (x + 2)/(x − 3).

例：在 y = (x + 2)/(x − 3) 中求 x。

y = (x + 2)/(x − 3)
y(x − 3) = x + 2
yx − 3y = x + 2
yx − x = 3y + 2
x(y − 1) = 3y + 2
x = (3y + 2)/(y − 1)

Let us examine each step. First multiply by x − 3 to remove the fraction. Then expand the left side. Next subtract x from both sides and add 3y to both sides so that all x terms are on the left. Factorise x out of the two left-hand terms. Finally divide by y − 1.

我们仔细看每一步。先乘以 x − 3 去掉分数。然后展开左边。接着两边同时减去 x 并加上 3y，把所有含 x 的项放到左边。从左边两项中提取公因式 x。最后除以 y − 1。

It is easy to make sign errors in this type of question. Always check the signs carefully when moving terms across the equals sign.

在这类问题中很容易出现符号错误。移动项过等号时，一定要仔细检查符号。

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7. Squares and Square Roots | 平方与平方根

If the variable you want is squared, you can use a square root to isolate it. Remember that the square root of a positive number has two values: one positive and one negative. In geometry and physics, where variables represent lengths or distances, we usually take only the positive root.

如果你想要的变量被平方了，你可以用平方根来隔离它。请记住正数的平方根有两个值：一正一负。在几何和物理

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