📚 Page 240 Problem 2: Solving Linear Equations in Geometry | 第240页第2题：几何中的一次方程求解
Many Cambridge KS3 learners meet problems like the one on page 240 where a geometric situation, such as a perimeter or an angle sum, must be expressed as a linear equation and then solved. This article reviews the key skills you need: balancing equations, expanding brackets, clearing fractions, and applying algebra to rectangles and angle facts. It also shows how to build equations from word problems, check your answers, and avoid common errors in checkpoint assessments.
许多剑桥 KS3 学生都会遇到第 240 页这类题目：必须把一个几何情境，比如周长或角度和，写成一个一次方程并求解。本文复习你需要的核心技能：平衡方程、展开括号、去分母，以及将代数应用到矩形和角度事实。文章还会展示如何从应用题建立方程、检查答案，并避免 checkpoint 测评中的常见错误。
1. What Is a Linear Equation? | 什么是一次方程
the power 1. It can be written in forms such as 2x + 3 = 11 or 5(n − 2) = 3n + 4. The word "linear" means the graph of the equation is a straight line.
一次方程含有一个未知数，通常写成 x 或 n，而且未知数只出现一次幂。它可以写成 2x + 3 = 11 或 5(n − 2) = 3n + 4 这样的形式。"线性"的意思是方程的图像是一条直线。
n use inverse operations to find the exact answer. The solution is the value that makes both sides of the equation equal. For example, x = 4 solves 2x + 3 = 11 because 2 × 4 + 3 = 11.
在 KS3 题目中，未知数通常表示一个长度、一个角度、一个费用或一个得分。你不需要用试错法猜数值；你可以使用逆运算求出准确答案。解就是让方程两边相等的值。例如 x = 4 是 2x + 3 = 11 的解，因为 2 × 4 + 3 = 11。
ecause the x term disappears. In geometry, the coefficient a often comes from a formula such as perimeter, while b and c come from the given measurements.
一次方程也可以整理成标准形式 ax + b = c，其中 a、b、c 是数，且 a ≠ 0。如果 a = 0，方程就不再是一次方程，因为 x 项消失了。在几何中，系数 a 常常来自周长等公式，而 b 和 c 来自题目给出的测量值。
2. The Balance Method | 平衡法
Think of an equation as a balance scale. The left side and right side must stay equal. Whatever operation you do to one side, you must do exactly the same operation to the other side. This is the golden rule for solving equations.
把方程想象成一台天平。左边和右边必须保持相等。你对一边做什么运算，就必须对另一边做完全相同的运算。这是解方程的金科玉律。
h the operations: undo addition or subtraction first, then undo multiplication or division.
例如，解 2x + 3 = 11 时，先从两边同时减去 3。这得到 2x = 8。然后两边同时除以 2，得到 x = 4。始终按运算的逆序操作：先抵消加减，再抵消乘除。
our solution is correct. For 2x + 3 = 11, substitute x = 4: 2 × 4 + 3 = 11, which is true.
你可以把解代回原方程来检查。如果两边得到相同的值，你的解就是正确的。对于 2x + 3 = 11，代入 x = 4：2 × 4 + 3 = 11，成立。
Common mistake: only changing one side of the equation.
常见错误：只改变方程的一边。
Common mistake: forgetting to divide the constant term as well as the x term.
常见错误：忘记常数项也要一起除，而不只是 x 项。
Exam tip: keep your equal signs aligned so each step is clear.
考试技巧：让等号对齐，使每一步都清晰。
3. Expanding Brackets in Linear Equations | 一次方程中的展开括号
en subtract 3n from both sides to get 2n − 10 = 4. Add 10 to both sides: 2n = 14. Divide by 2: n = 7.
几何中的许多一次方程含有括号。使用分配律：a(b + c) = ab + ac。例如，5(n − 2) = 3n + 4 展开为 5n − 10 = 3n + 4。然后两边同时减去 3n，得到 2n − 10 = 4。两边同时加 10：2n = 14。除以 2：n = 7。
kets are subtracted, be careful with signs. For example, −2(x + 3) = −2x − 6, not −2x + 6.
检查：5(7 − 2) = 5 × 5 = 25，而 3 × 7 + 4 = 21 + 4 = 25。两边相等，所以 n = 7 正确。当括号前是减号时，要特别小心符号。例如，−2(x + 3) = −2x − 6，而不是 −2x + 6。
idth is x, then P = 2(x + 3 + x) = 2(2x + 3) = 4x + 6. Setting this equal to a given perimeter creates a linear equation.
在几何中，括号经常出现在周长公式里。长为 l、宽为 w 的矩形，周长 P = 2(l + w)。如果长是 x + 3，宽是 x，那么 P = 2(x + 3 + x) = 2(2x + 3) = 4x + 6。让它等于题目给出的周长，就得到一个一次方程。
4. Clearing Fractions and Decimals | 去分母与去小数
− 2) ÷ 2. Multiply both sides by 6: 2(x + 1) = 3(x − 2). Expand: 2x + 2 = 3x − 6. Rearrange: 2 = x − 6, so x = 8.
含分数的方程可以通过两边同时乘以分母的最小公倍数来解。例如，解 (x + 1) ÷ 3 = (x − 2) ÷ 2。两边同时乘以 6：2(x + 1) = 3(x − 2)。展开：2x + 2 = 3x − 6。整理：2 = x − 6，所以 x = 8。
.7 can be multiplied by 10 to give 3x + 12 = 27. Subtract 12: 3x = 15. Divide by 3: x = 5.
小数可以用类似的方法清除，两边同时乘以 10、100 或其他 10 的幂。例如，0.3x + 1.2 = 2.7 可以两边乘以 10，得到 3x + 12 = 27。减去 12：3x = 15。除以 3：x = 5。
In geometry, fractions often come from ratios, scale factors, or angle expressions such as (2x + 10) ÷ 5. Clearing the fraction before solving usually makes the equation easier to manage.
在几何中，分数常常来自比例、比例尺或角度表达式，例如 (2x + 10) ÷ 5。在求解前先去分母通常会让方程更容易处理。
Multiply every term by the same number.
每一项都要乘以同一个数。
Use brackets when multiplying an expression with more than one term.
当乘以含有多项的表达式时，要加括号。
Check that no denominator equals zero.
检查分母是否可能为零。
5. Geometry 1: Perimeter and Rectangle Problems | 几何一：周长与矩形问题
uation. For example, a rectangle is 3 cm longer than it is wide, and its perimeter is 26 cm. Let the width be w. Then the length is w + 3. The perimeter is 2(w + w + 3) = 26.
周长问题是线性方程的常见来源。对矩形来说，P = 2(l + w)。如果长用宽表示，就可以建立方程。例如，一个矩形比宽长 3 cm，周长是 26 cm。设宽为 w。那么长是 w + 3。周长是 2(w + w + 3) = 26。
= 5. The width is 5 cm, and the length is 5 + 3 = 8 cm. Check: 2(5 + 8) = 2 × 13 = 26 cm.
化简：2(2w + 3) = 26，所以 4w + 6 = 26。减去 6：4w = 20。除以 4：w = 5。宽是 5 cm，长是 5 + 3 = 8 cm。检查：2(5 + 8) = 2 × 13 = 26 cm。
he sides are algebraic expressions, add them and set the sum equal to the given perimeter.
其他周长图形包括三角形，P = a + b + c，以及正方形，P = 4s。如果边长是代数表达式，就把它们相加，并让总和等于给定的周长。
| Shape | 图形 | Perimeter formula | 周长公式 |
| Rectangle | 矩形 | P = 2(l + w) | P = 2(l + w) |
| Square | 正方形 | P = 4s | P = 4s |
| Triangle | 三角形 | P = a + b + c | P = a + b + c |
6. Geometry 2: Angles on a Straight Line and at a Point | 几何二：直线上角与点周围角
Angle facts give you ready-made equations. Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal. These facts turn angle diagrams into linear equations.
角度事实为你提供现成的方程。直线上的角相加等于 180°。点周围的角相加等于 360°。对顶角相等。这些事实把角度图变成一次方程。
act 40: 5x = 140. Divide by 5: x = 28. The angles are 3 × 28 + 10 = 94° and 2 × 28 + 30 = 86°. Check: 94 + 86 = 180.
例如，直线上的两个角是 (3x + 10)° 和 (2x + 30)°。因为它们相加等于 180°，我们写出 3x + 10 + 2x + 30 = 180。化简：5x + 40 = 180。减去 40：5x = 140。除以 5：x = 28。两个角分别是 3 × 28 + 10 = 94° 和 2 × 28 + 30 = 86°。检查：94 + 86 = 180。
site angle is (2x + 25)°, then 4x − 5 = 2x + 25, so 2x = 30 and x = 15. Each angle is 55°.
如果两个角是对顶角，就让它们相等。例如，一个角是 (4x − 5)°，它的对顶角是 (2x + 25)°，那么 4x − 5 = 2x + 25，所以 2x = 30，x = 15。每个角都是 55°。
Always check that your final angles are sensible: acute angles are less than 90°, obtuse angles are between 90° and 180°, and angles on a straight line must sum to exactly 180°.
始终检查最终角度是否合理：锐角小于 90°，钝角在 90° 到 180° 之间，直线上的角必须正好相加为 180°。
7. Geometry 3: Triangles and Angle Sums | 几何三：三角形与角度和
The interior angles of a triangle add up to 180°. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. These facts allow you to solve for unknown angles using linear equations.
三角形的内角相加等于 180°。三角形的一个外角等于与它不相邻的两个内角之和。这些事实让你可以用一次方程求未知角。
ivide by 4: x = 42.5. The angles are 42.5°, 62.5°, and 75°. Check: 42.5 + 62.5 + 75 = 180.
例如，三角形的三个角是 x°、(x + 20)° 和 (2x − 10)°。写出 x + (x + 20) + (2x − 10) = 180。化简：4x + 10 = 180。减去 10：4x = 170。除以 4：x = 42.5。三个角是 42.5°、62.5° 和 75°。检查：42.5 + 62.5 + 75 = 180。
0°. If the base angles are given as (3x + 5)°, you can set up 2(3x + 5) + top angle = 180.
在等腰三角形中，底角相等。如果顶角是 40°，那么每个底角是 (180 − 40) ÷ 2 = 70°。如果底角表示为 (3x + 5)°，你可以建立 2(3x + 5) + 顶角 = 180。
For quadrilaterals, the interior angles add up to 360°. This is useful for trapeziums, parallelograms, and other four-sided shapes in KS3 geometry questions.
对于四边形，内角相加等于 360°。这对 KS3 几何题中的梯形、平行四边形和其他四边形很有用。
