KS3 Year 8 Maths: Solving Linear Equations and Straight-Line Graphs — 七年级数学：解一元一次方程与直线图像

一、什么是一元一次方程？平衡秤上的等式 | What Is a Linear Equation? An Equality on a Balance Scale

在七年级和八年级的数学课上，一元一次方程是代数学习的第一个核心工具。它的英文名称是 linear equation in one variable，因为它只含有一个未知数（通常用字母 x 表示），并且这个未知数的最高次数是 1。形如 2x + 5 = 13、3(x – 2) = 9 这样的式子，都是一元一次方程。

In Year 7 and Year 8 mathematics, the linear equation in one variable is the first core tool of algebra. It gets its name because it contains only one unknown (usually written as the letter x), and that unknown is raised to the power 1 at most. Expressions such as 2x + 5 = 13 and 3(x – 2) = 9 are both linear equations in one variable.

理解方程最好的办法，是把它想象成一台两边保持平衡的天平。等号左右两边各放一个秤盘，左边放 2x + 5，右边放 13。只要天平平衡，两边就相等。我们解方程的目标，就是通过一系列”同时操作”找出让天平保持平衡的那个 x 的值。

The best way to understand an equation is to picture a balance scale that stays level. The left pan holds 2x + 5 and the right pan holds 13. As long as the scale balances, the two sides are equal. Our goal when solving an equation is to find the value of x that keeps the scale balanced, using a series of operations performed on both sides at once.

要记住一条黄金法则：对等号一边做的任何事，必须对另一边做完全相同的操作。这个原则叫做”平衡原则”（balance method），是后面所有解题步骤的基础。

Remember one golden rule: whatever you do to one side of the equals sign, you must do exactly the same thing to the other side. This principle is called the balance method, and it underpins every solving step that follows.

二、平衡法解题：逆向操作与等号两边同加同减 | Solving by the Balance Method: Inverse Operations and Doing the Same to Both Sides

解方程的核心思路，是把未知数 x 单独留在等号一边。为了做到这一点，我们使用”逆向操作”：加法对应减法，减法对应加法，乘法对应除法，除法对应乘法。每一步都要在等号两边同时进行。

The core idea of solving an equation is to leave the unknown x on its own on one side of the equals sign. To do this we use inverse operations: addition is undone by subtraction, subtraction by addition, multiplication by division, and division by multiplication. Every step must be applied to both sides at the same time.

看一个最简单的例子：x + 7 = 15。因为 x 被加了 7，我们要在两边同时减去 7。左边变成 x + 7 – 7 = x，右边变成 15 – 7 = 8，于是得到 x = 8。检验一下：8 + 7 = 15，正确。

Take the simplest example: x + 7 = 15. Because 7 has been added to x, we subtract 7 from both sides. The left side becomes x + 7 – 7 = x, and the right side becomes 15 – 7 = 8, giving x = 8. Check the answer: 8 + 7 = 15, which is correct.

再看一个减法的例子：x – 5 = 9。x 被减去了 5，所以我们要在两边同时加上 5。得到 x = 14。检验：14 – 5 = 9，正确。

Now try a subtraction example: x – 5 = 9. Since 5 has been subtracted from x, we add 5 to both sides. We get x = 14. Check: 14 – 5 = 9, which is correct.

乘法的情况稍微不同。例如 4x = 28 表示”x 乘以 4 等于 28″。要撤销乘以 4，就要在两边同时除以 4，得到 x = 28 ÷ 4 = 7。检验：4 × 7 = 28，正确。

Multiplication works slightly differently. For example, 4x = 28 means “x multiplied by 4 equals 28”. To undo the multiplication by 4, we divide both sides by 4, giving x = 28 ÷ 4 = 7. Check: 4 × 7 = 28, correct.

最后是除法：x ÷ 3 = 6 表示 x 被 3 除了。要撤销除以 3，就在两边同时乘以 3，得到 x = 18。检验：18 ÷ 3 = 6，正确。这四种基本类型覆盖了所有一元一次方程的解法。

Finally, division: x ÷ 3 = 6 means x has been divided by 3. To undo the division by 3, multiply both sides by 3, giving x = 18. Check: 18 ÷ 3 = 6, correct. These four basic types cover the solution of every linear equation in one variable.

三、两步方程：先处理加减，再处理乘除 | Two-Step Equations: Handle Addition or Subtraction Before Multiplication or Division

大多数方程需要两步才能解出。例如 2x + 5 = 13，这里 x 先被乘以 2，再加上 5。解这类方程时，必须把顺序反过来：先撤销”加 5″，再撤销”乘 2″。

Most equations need two steps to solve. Take 2x + 5 = 13, where x is first multiplied by 2 and then 5 is added. To solve this kind of equation, the order must be reversed: undo the “+5” first, then undo the “times 2”.

第一步：两边同时减去 5。左边 2x + 5 – 5 = 2x，右边 13 – 5 = 8，得到 2x = 8。第二步：两边同时除以 2，得到 x = 4。完整检验：2 × 4 + 5 = 8 + 5 = 13，正确。

Step one: subtract 5 from both sides. The left side becomes 2x + 5 – 5 = 2x, and the right side becomes 13 – 5 = 8, giving 2x = 8. Step two: divide both sides by 2 to get x = 4. Full check: 2 × 4 + 5 = 8 + 5 = 13, correct.

这里有一个必须牢记的顺序规则：先撤销最外层的加减运算，再撤销乘除运算。很多学生一上来就想除以 2，得到 x + 2.5 = 6.5，虽然也能继续算，但会引入讨厌的小数，更容易出错。先减再加、先除再乘，永远先处理加减。

There is an ordering rule you must remember: undo the outer addition or subtraction first, then undo the multiplication or division. Many students try to divide by 2 straight away, getting x + 2.5 = 6.5, which can still be solved but introduces awkward decimals and invites mistakes. Always deal with the addition or subtraction before the multiplication or division.

另一个常见类型是 3x – 8 = 7。第一步两边同时加 8，得到 3x = 15；第二步两边除以 3，得到 x = 5。检验：3 × 5 – 8 = 15 – 8 = 7，正确。

Another common type is 3x – 8 = 7. Step one: add 8 to both sides to get 3x = 15. Step two: divide both sides by 3 to get x = 5. Check: 3 × 5 – 8 = 15 – 8 = 7, correct.

四、带括号的方程：先用乘法分配律展开 | Equations with Brackets: Expand First Using the Distributive Law

当方程里出现括号时，例如 3(x – 2) = 9，第一步通常是”展开括号”。括号前的数字要乘到括号里的每一项：3(x – 2) = 3x – 6。这个规则叫做乘法分配律（distributive law）。

When a bracket appears in an equation, such as 3(x – 2) = 9, the first step is usually to expand the bracket. The number in front multiplies every term inside: 3(x – 2) = 3x – 6. This rule is called the distributive law.

于是 3(x – 2) = 9 变成 3x – 6 = 9。接着是熟悉的两步：两边加 6 得到 3x = 15，两边除以 3 得到 x = 5。检验：3 × (5 – 2) = 3 × 3 = 9，正确。

So 3(x – 2) = 9 becomes 3x – 6 = 9. Then come the familiar two steps: add 6 to both sides to get 3x = 15, and divide by 3 to get x = 5. Check: 3 × (5 – 2) = 3 × 3 = 9, correct.

负号要特别小心。例如 2(3x + 4) – 5 = 21，先展开 2(3x + 4) = 6x + 8，方程变成 6x + 8 – 5 = 21，即 6x + 3 = 21。两边减 3 得 6x = 18，两边除以 6 得 x = 3。检验：2 × (9 + 4) – 5 = 26 – 5 = 21，正确。

Be especially careful with negative signs. For example, 2(3x + 4) – 5 = 21. First expand 2(3x + 4) = 6x + 8, so the equation becomes 6x + 8 – 5 = 21, which is 6x + 3 = 21. Subtract 3 from both sides to get 6x = 18, then divide by 6 to get x = 3. Check: 2 × (9 + 4) – 5 = 26 – 5 = 21, correct.

括号前面是减号时，展开后括号里每一项的符号都要反过来。例如 10 – 2(x + 1) = 2 中，-2(x + 1) = -2x – 2，所以方程变成 10 – 2x – 2 = 2，即 8 – 2x = 2。两边减 8 得 -2x = -6，两边除以 -2 得 x = 3。

When a minus sign sits in front of a bracket, every term inside flips sign when expanded. For example, in 10 – 2(x + 1) = 2, we have -2(x + 1) = -2x – 2, so the equation becomes 10 – 2x – 2 = 2, that is 8 – 2x = 2. Subtract 8 from both sides to get -2x = -6, then divide by -2 to get x = 3.

五、含分数的方程：去分母让式子变简单 | Equations with Fractions: Clear the Denominators to Simplify

含分数的方程看起来吓人，但只要记住一个技巧：先”去分母”。做法是找到所有分母的最小公倍数（LCM），然后把方程两边同时乘以这个数，分数就消失了。

Equations with fractions can look intimidating, but there is one trick to remember: clear the denominators first. Find the lowest common multiple (LCM) of all the denominators, then multiply both sides of the equation by that number. The fractions disappear.

例如 x/3 + 1 = 5。分母是 3，两边同时乘以 3：x + 3 = 15。两边减 3 得 x = 12。检验：12 ÷ 3 + 1 = 4 + 1 = 5，正确。

For example, x/3 + 1 = 5. The denominator is 3, so multiply both sides by 3: x + 3 = 15. Subtract 3 from both sides to get x = 12. Check: 12 ÷ 3 + 1 = 4 + 1 = 5, correct.

更复杂一点的例子：x/2 = x/3 + 2。分母有 2 和 3，最小公倍数是 6。两边同时乘以 6：6 × x/2 = 6 × x/3 + 6 × 2，即 3x = 2x + 12。两边减 2x 得 x = 12。检验：12/2 = 6，12/3 + 2 = 4 + 2 = 6，两边相等，正确。

A slightly harder example: x/2 = x/3 + 2. The denominators are 2 and 3, whose LCM is 6. Multiply both sides by 6: 6 × x/2 = 6 × x/3 + 6 × 2, giving 3x = 2x + 12. Subtract 2x from both sides to get x = 12. Check: 12/2 = 6, and 12/3 + 2 = 4 + 2 = 6. Both sides match, correct.

去分母时务必把”整项”都乘到。像 x/2 + 3 = 5 乘以 2 之后，3 也要乘以 2，变成 x + 6 = 10，而不是 x + 3 = 10。忘记乘常数项是最常见的错误之一。

When clearing denominators, make sure to multiply every single term. In x/2 + 3 = 5, after multiplying by 2, the 3 must also be multiplied by 2, giving x + 6 = 10, not x + 3 = 10. Forgetting to multiply the constant term is one of the most common mistakes.

六、未知数在等号两边：把所有 x 移到同一边 | Unknowns on Both Sides: Collect All the x Terms on One Side

有些方程等号两边都含有未知数，例如 5x – 2 = 3x + 6。解这类方程的原则是：把所有含 x 的项移到一边，把所有数字移到另一边。移动项时要改变符号。

Some equations have unknowns on both sides, such as 5x – 2 = 3x + 6. The principle for solving them is to collect all the x terms on one side and all the numbers on the other. When a term moves across the equals sign, its sign changes.

把 3x 移到左边（变号成 -3x），把 -2 移到右边（变号成 +2）：5x – 3x = 6 + 2，即 2x = 8，所以 x = 4。检验：5 × 4 – 2 = 18，3 × 4 + 6 = 18，两边相等，正确。

Move the 3x to the left (its sign flips to -3x), and move the -2 to the right (its sign flips to +2): 5x – 3x = 6 + 2, giving 2x = 8, so x = 4. Check: 5 × 4 – 2 = 18 and 3 × 4 + 6 = 18. Both sides match, correct.

一个关键技巧：如果 x 前面的系数变成负数，比如 -2x = 6，最简单的方法就是两边同时除以那个负数，得到 x = -3。或者也可以先在两边同时加 2x，把负系数移到另一边变成正数。

A key tip: if the coefficient of x turns negative, say -2x = 6, the simplest move is to divide both sides by that negative number, giving x = -3. Alternatively, add 2x to both sides to move the negative coefficient to the other side where it becomes positive.

七、文字应用题：把中文句子翻译成方程 | Word Problems: Translating Sentences into Equations

文字应用题是考试的重头戏，考察的是把语言翻译成数学的能力。解题分四步：读题并设未知数、把条件写成方程、解方程、把答案代回原题检验是否合理。

Word problems are a major part of exams, testing your ability to translate language into mathematics. Solving them follows four steps: read the problem and define the unknown, write the conditions as an equation, solve the equation, and substitute the answer back to check that it makes sense.

常见的关键词有：”比…多”表示加法，”比…少”表示减法，”的几倍”表示乘法，”平均分”表示除法，”等于””一共””总计”表示等号。掌握这些关键词，就能快速把句子变成式子。

Common keywords include: “more than” means addition, “less than” means subtraction, “times as many” means multiplication, “shared equally” means division, and “equals”, “altogether” or “in total” mark the equals sign. Master these keywords and you can turn sentences into expressions quickly.

例题：一个数的 3 倍加上 7 等于 25，求这个数。设这个数为 x，则 3x + 7 = 25。两边减 7 得 3x = 18，两边除以 3 得 x = 6。答：这个数是 6。检验：3 × 6 + 7 = 25，正确。

Example: three times a number plus 7 equals 25. Find the number. Let the number be x, so 3x + 7 = 25. Subtract 7 from both sides to get 3x = 18, then divide by 3 to get x = 6. Answer: the number is 6. Check: 3 × 6 + 7 = 25, correct.

更复杂一点的例题：长是宽的 2 倍，长方形的周长是 24，求长和宽。设宽为 x，则长为 2x。周长 = 2 × (长 + 宽) = 2 × (2x + x) = 6x。所以 6x = 24，x = 4。答：宽 4，长 8。

A slightly harder example: the length is twice the width, and the perimeter of the rectangle is 24. Find the length and width. Let the width be x, so the length is 2x. Perimeter = 2 × (length + width) = 2 × (2x + x) = 6x. So 6x = 24, giving x = 4. Answer: width 4, length 8.

八、坐标平面：x 轴、y 轴与点的位置 | The Coordinate Plane: The x-Axis, y-Axis and the Position of Points

方程和图像是一对亲密伙伴。要画直线图像，先要熟悉坐标平面。坐标平面由两条垂直的数轴组成：水平的叫 x 轴，竖直的叫 y 轴，它们的交点是原点 (0, 0)。

Equations and graphs are close partners. Before drawing straight-line graphs, get comfortable with the coordinate plane. It is made of two perpendicular number lines: the horizontal one is the x-axis, the vertical one is the y-axis, and their crossing point is the origin (0, 0).

一个点的位置用一对有序数 (x, y) 表示。x 是横坐标，表示沿水平方向离原点多远；y 是纵坐标，表示沿竖直方向离原点多远。例如点 (3, 2) 表示”向右走 3，再向上走 2″。顺序绝对不能颠倒。

A point’s position is given by an ordered pair (x, y). The x-coordinate tells how far horizontally from the origin, and the y-coordinate tells how far vertically. For example, the point (3, 2) means “go right 3, then up 2”. The order can never be swapped.

四个象限（quadrants）是考试常考点：右上为第一象限（x 和 y 都为正），左上为第二象限（x 为负，y 为正），左下为第三象限（都为负），右下为第四象限（x 为正，y 为负）。

The four quadrants are a common exam focus: the top-right is the first quadrant (both x and y positive), top-left is the second (x negative, y positive), bottom-left is the third (both negative), and bottom-right is the fourth (x positive, y negative).

九、直线的方程：y = mx + c 中 m 与 c 的含义 | The Equation of a Straight Line: What m and c Mean in y = mx + c

所有直线都可以写成 y = mx + c 的形式。其中 m 是斜率（gradient），表示直线的倾斜程度；c 是纵截距（y-intercept），表示直线与 y 轴相交的位置。这个式子就是直线的”身份证”。

Every straight line can be written in the form y = mx + c. Here m is the gradient, which measures how steep the line is, and c is the y-intercept, the place where the line crosses the y-axis. This formula is the “identity card” of a straight line.

斜率 m 的计算方法是：竖直变化量除以水平变化量，也就是 m = 上升/前进（rise over run）。例如 m = 2 表示”每向右走 1 格，就向上走 2 格”；m = -1 表示”每向右走 1 格，就向下走 1 格”。

The gradient m is calculated as the vertical change divided by the horizontal change, that is m = rise over run. For example, m = 2 means “for every 1 unit to the right, go up 2 units”, while m = -1 means “for every 1 unit to the right, go down 1 unit”.

纵截距 c 直接告诉你直线在哪里穿过 y 轴。y = 2x + 3 这条线在点 (0, 3) 处穿过 y 轴，因为当 x = 0 时，y = 2 × 0 + 3 = 3。所以 c = 3。

The y-intercept c tells you exactly where the line crosses the y-axis. The line y = 2x + 3 crosses the y-axis at (0, 3), because when x = 0, y = 2 × 0 + 3 = 3. So c = 3.

十、画直线图像：描点法的四步流程 | Plotting a Straight Line: The Four-Step Table Method

画一条直线只需要两个点，但通常我们描三个点来确保没有算错。描点法分四步：第一步，选几个 x 值（建议 -2、-1、0、1、2）；第二步，把每个 x 代入方程算出对应的 y 值；第三步，在坐标平面上标出这些点；第四步，用直尺连成一条直线。

A straight line needs only two points, but we usually plot three to guard against mistakes. The table method has four steps: first, choose several x-values (try -2, -1, 0, 1, 2); second, substitute each x into the equation to find the matching y-value; third, mark the points on the coordinate plane; fourth, join them with a ruler into a straight line.

以 y = 2x + 1 为例：当 x = -1 时 y = -1；当 x = 0 时 y = 1；当 x = 1 时 y = 3；当 x = 2 时 y = 5。得到四个点 (-1, -1)、(0, 1)、(1, 3)、(2, 5)，它们整齐地排在一条直线上。

Take y = 2x + 1: when x = -1, y = -1; when x = 0, y = 1; when x = 1, y = 3; when x = 2, y = 5. This gives four points (-1, -1), (0, 1), (1, 3) and (2, 5), all sitting neatly on one straight line.

描点后检查一下：如果三个点不在同一条直线上，说明至少有一个点算错了，要回头重新代入检验。直线图像永远是直的，这是它名字的来源，也是检查错误的有力武器。

After plotting, check: if the three points do not line up on one straight line, at least one of them was calculated wrongly, so go back and substitute again. A straight-line graph is always straight, which is where it gets its name and is also a powerful way to catch errors.

十一、从图像读信息：根据直线写方程 | Reading Information from a Graph: Writing the Equation from a Line

反过来，给你一条已经画好的直线，你也要能写出它的方程 y = mx + c。分两步：先找 c，也就是直线与 y 轴的交点；再找 m，也就是任取两点计算斜率。

The reverse skill is just as important: given a line already drawn, write its equation y = mx + c. Do it in two steps: first find c, the point where the line crosses the y-axis; then find m, the gradient calculated from any two points on the line.

例如一条直线穿过点 (0, 2) 和 (1, 5)。它与 y 轴交于 (0, 2)，所以 c = 2。计算斜率：两点之间 x 增加 1，y 增加 3，所以 m = 3 ÷ 1 = 3。于是直线的方程是 y = 3x + 2。

For example, a line passes through (0, 2) and (1, 5). It crosses the y-axis at (0, 2), so c = 2. To find the gradient: between the two points, x increases by 1 and y increases by 3, so m = 3 ÷ 1 = 3. The equation of the line is therefore y = 3x + 2.

斜率的正负决定直线的走向：m 为正时直线从左下向右上倾斜（递增），m 为负时从左上向右下倾斜（递减），m = 0 时是水平直线（如 y = 4）。竖直直线的方程写不成 y = mx + c，它要写成 x = k 的形式。

The sign of the gradient decides the line’s direction: when m is positive the line rises from bottom-left to top-right (increasing), when m is negative it falls from top-left to bottom-right (decreasing), and when m = 0 it is horizontal (such as y = 4). A vertical line cannot be written as y = mx + c; it must be written as x = k.

十二、方程与图像的联系：交点就是方程的解 | Connecting Equations and Graphs: The Intersection Is the Solution

方程和直线图像最漂亮的联系是：方程的解，恰好就是图像与 x 轴的交点（或者说图像在某个 y 值处对应的 x）。例如 y = 2x – 4 与 x 轴交于点 (2, 0)，那么方程 2x – 4 = 0 的解就是 x = 2。

The most beautiful link between equations and graphs is this: the solution of an equation is exactly where the graph meets the x-axis (or the x-value the graph takes at a given y). For example, y = 2x – 4 crosses the x-axis at (2, 0), so the solution of 2x – 4 = 0 is x = 2.

同样的，两条直线的交点可以同时满足两个方程。例如 y = 2x 和 y = x + 3 的交点，就是使 2x = x + 3 成立的 x 值。解这个方程得 x = 3，代入任一式得 y = 6，所以交点是 (3, 6)。

Likewise, the intersection of two lines satisfies both equations at once. For example, the crossing point of y = 2x and y = x + 3 is the x-value that makes 2x = x + 3 true. Solving gives x = 3, and substituting into either equation gives y = 6, so the intersection is (3, 6).

这个”图像与方程一一对应”的思想，是八年级数学里最重要的抽象飞跃。它把代数（字母和等式）和几何（点、线和形状）连在了一起，为九年级和更高年级的函数学习打下基础。

This idea that graphs and equations correspond one to one is the most important abstract leap in Year 8 mathematics. It connects algebra (letters and equations) with geometry (points, lines and shapes), laying the foundation for the study of functions in Year 9 and beyond.

十三、常见错误与考试技巧：验算、写步骤、看清负号 | Common Mistakes and Exam Tips: Check, Show Working, and Watch the Signs

考试中最常见的失分点有三个。第一是”跳步”：直接心算出答案却没有写过程，一旦算错就全扣。第二是”负号错误”：移项或去括号时忘记变号。第三是”不验算”：解完就把答案代入原方程验证一遍，能立刻发现绝大多数错误。

Three mistakes cost the most marks in exams. The first is skipping steps: working the answer out mentally without showing working means a single slip loses everything. The second is sign errors: forgetting to flip a sign when moving a term or expanding a bracket. The third is not checking: substituting your answer back into the original equation catches the vast majority of errors instantly.

考试技巧：每题都写清楚”两边同时做什么”，让阅卷老师能看到你的思路；负号用彩色笔圈出来提醒自己；遇到分数先通分或去分母；最后留一分钟把答案代回原式检验。

Exam tips: write clearly “what you did to both sides” for every question so the examiner can follow your reasoning; circle negative signs in a different colour as a reminder; clear denominators whenever fractions appear; and leave a minute at the end to substitute each answer back into the original equation.

把方程和直线图像结合起来复习，效率最高。解方程时想想图像长什么样，画图时想想这条线对应哪个方程。两个方向都熟练了，这一章就真正过关了。

The most efficient revision combines equations with their graphs. When solving an equation, picture what its graph looks like; when drawing a graph, think about which equation it represents. Once you are fluent in both directions, you have truly mastered this topic.

十四、两点求斜率：上升除以前进的精确计算 | Finding the Gradient from Two Points: Rise over Run in Detail

给一条直线上的两个点 (x1, y1) 和 (x2, y2)，斜率可以用公式 m = (y2 – y1) ÷ (x2 – x1) 精确算出。这个公式其实就是”竖直变化量除以水平变化量”，是八年级最常用也最好记的公式之一。

Given two points (x1, y1) and (x2, y2) on a line, the gradient can be found exactly with the formula m = (y2 – y1) ÷ (x2 – x1). This is just “vertical change divided by horizontal change”, and it is one of the most useful and memorable formulas in Year 8.

例题：直线经过 (1, 3) 和 (4, 9) 两点，求斜率。代入公式：m = (9 – 3) ÷ (4 – 1) = 6 ÷ 3 = 2。所以斜率是 2。注意分子和分母的顺序要和两个点的坐标顺序保持一致。

Example: a line passes through (1, 3) and (4, 9). Find its gradient. Substitute into the formula: m = (9 – 3) ÷ (4 – 1) = 6 ÷ 3 = 2. So the gradient is 2. Keep the numerator and denominator in the same point order to avoid mistakes.

再举一个斜率为负的例子：直线经过 (2, 7) 和 (5, 1)。m = (1 – 7) ÷ (5 – 2) = -6 ÷ 3 = -2。负号表示直线从左向右是下降的。用图像画出来验证，会发现两点确实连成一条向下的直线。

Now a negative-gradient example: a line passes through (2, 7) and (5, 1). Then m = (1 – 7) ÷ (5 – 2) = -6 ÷ 3 = -2. The negative sign tells us the line falls as we move left to right. Plot the points to confirm that they join into a downward-sloping line.

一旦算出斜率 m，再结合直线与 y 轴的交点得到 c，就能写出完整的直线方程。先用两点求 m，再代入其中一点解出 c，是”已知两点求直线方程”问题的标准三步法。

Once you have the gradient m, combine it with the y-intercept c to write the full equation of the line. Finding m from two points first, then substituting one point to solve for c, is the standard three-step method for “find the equation given two points” questions.

Summary | 总结

一元一次方程是代数的基石：它只含一个未知数，最高次数为 1，通过”对两边做相同操作”的平衡法求解。解方程的步骤永远是先展开括号、再去分母、再移项合并、最后解出未知数，每一步都遵循逆向操作的逻辑。

The linear equation in one variable is the cornerstone of algebra: it contains one unknown raised to the power 1, and it is solved by the balance method of doing the same thing to both sides. The solving order is always expand brackets, clear fractions, collect like terms, then solve for the unknown, with each step following the logic of inverse operations.

直线图像由方程 y = mx + c 完全决定，其中 m 是斜率、c 是纵截距。描点法把代数方程变成可视的直线，而反过来，从一条直线也能读出它的方程。方程的解对应图像与 x 轴的交点，两条直线的交点同时满足两个方程。

A straight-line graph is completely determined by its equation y = mx + c, where m is the gradient and c is the y-intercept. The table method turns an algebraic equation into a visible straight line, and in reverse, a line’s equation can be read straight off the graph. The solution of an equation matches where its graph meets the x-axis, and the intersection of two lines satisfies both equations at once.

掌握这一章的关键在于三点：理解平衡原则，熟练逆向操作，以及建立方程与图像之间的双向联系。勤加练习、认真验算，一元一次方程与直线图像一定能成为你的强项。

The key to mastering this topic is threefold: understand the balance principle, become fluent with inverse operations, and build a two-way connection between equations and graphs. With steady practice and careful checking, linear equations and straight-line graphs will become one of your strongest areas.

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