📚 Unit 06 Linear Equations and Graphs | 第6单元 线性方程与图像
te plane. Every linear equation in one unknown can be written in the form ax + b = c, and every straight line on a graph can be written as y = mx + c; in fact these are the same relationship seen from two different angles. This revision guide works through the unit skill by skill: equations with brackets, fractions and unknowns on both sides, rearranging formulae, solving inequalities, plotting graphs, reading gradients and intercepts, and using straight lines to model real situations. Each section gives worked examples, exam-style wording and the checking habits that stop careless marks from disappearing.
第6单元把9年级代数的两大核心内容结合在一起：解线性方程，以及在坐标平面上描述直线。一元线性方程都可以写成 ax + b = c 的形式，图像上的每条直线也都可以写成 y = mx + c 的形式；事实上，这两者只是同一个关系从不同角度呈现出来的样子。本复习指南按技能逐项梳理本单元内容：含括号、含分数、未知数出现在两边的方程，公式变形，解不等式，绘制图像，读取斜率与截距，并利用直线为现实情境建模。每一节都配有例题、考试常见表述，以及能避免粗心丢分的检验习惯。
1. What a Linear Equation Is | 什么是线性方程
denominators. Its general form is ax + b = c, where x is the unknown and a, b, c are constants with a ≠ 0. Solving the equation means finding every value of x that makes the left-hand side equal the right-hand side, and that value is called the solution or root.
线性方程是指未知数只出现一次方的方程；方程中不含未知数的平方、立方、根号，未知数也不出现在分母里。它的一般形式是 ax + b = c，其中 x 是未知数，a、b、c 是常数且 a ≠ 0。解方程就是找出所有能让左边等于右边的 x 值，这个值叫做方程的解（或根）。
The word "linear" is a hint about the graph: if you plot both sides of the equation as two straight lines, the solution is the x-value of the point where they cross. A linear equation in one unknown normally has exactly one solution, but two special cases exist and examiners like to test them.
"线性"这个词暗示了图像的特征：如果把等式两边分别画成两条直线，方程的解就是它们交点的横坐标。一元线性方程通常恰好有一个解，但有两种特殊情况，出题人很喜欢考查。
n: if simplifying gives something never true, such as 0 = 5, the equation has no solution.
情形一——恒等式：如果化简后得到恒成立的式子，例如 0 = 0，那么方程有无穷多个解。情形二——矛盾式：如果化简后得到永不成立的式子，例如 0 = 5，那么方程无解。
ntrast, x + 1 = x + 5 simplifies to 1 = 5, which is false, so there is no solution at all.
例如，2(x + 3) = 2x + 6 展开后得到 2x + 6 = 2x + 6，这是恒等式，所以任何 x 都成立。相反，x + 1 = x + 5 化简后得到 1 = 5，这是错的，所以方程完全无解。
ax + b = c (a ≠ 0) → x = (c − b) ÷ a
2. Solving Two-Step Equations | 解两步方程
Most equations in this unit need two inverse operations, and the safest order is to undo addition or subtraction first, then undo multiplication or division. Always apply the same operation to both sides so the equation stays balanced, and keep one equal sign under the other as you work down the page.
本单元大多数方程需要两次逆运算，最稳妥的顺序是先消去加减，再消去乘除。每一步都要对等式两边施加相同的运算，使等式保持平衡，并且书写时把等号上下对齐。
ing: 5 × 3 + 7 = 15 + 7 = 22, which matches the right-hand side, so the answer is correct.
例题：解 5x + 7 = 22。两边同时减去 7，得到 5x = 15；再两边同时除以 5，得到 x = 3。检验：把 x = 3 代回，5 × 3 + 7 = 15 + 7 = 22，与右边一致，所以答案正确。
5x + 7 = 22 → 5x = 15 → x = 3
- Step 1: Identify the operations acting on x, working outwards from x.
- 第一步：从 x 出发向外观察，找出作用在 x 上的运算。
- Step 2: Undo addition or subtraction on both sides.
- 第二步：在等式两边同时消去加法或减法。
- Step 3: Undo multiplication or division on both sides.
- 第三步：在等式两边同时消去乘法或除法。
- Step 4: Substitute your answer back into the original equation to check.
- 第四步：把答案代回原方程进行检验。
is exact, whereas 2.75 is only an approximation of the same value written in another form.
如果答案是一个分数，除非题目要求写成小数，否则应保留精确分数形式。例如 4x − 1 = 10 得到 4x = 11，x = 11 ÷ 4，这是精确值；而 2.75 只是同一数值的另一种写法。
3. Equations with Brackets and Fractions | 含括号和分数的方程
e resulting linear equation. Expanding first avoids sign errors and makes every term visible, which matters when marks are given for method.
当方程含有括号时，先用分配律 a(b + c) = ab + ac 展开括号，再解得到的线性方程。先展开可以避免符号错误，并让每一项都清晰可见，这在按步骤给分时非常重要。
both sides to get 3x = 15, then divide by 3 to get x = 5. Check: 3 × (5 + 4) = 3 × 9 = 27.
例题：3(x + 4) = 27 展开后得到 3x + 12 = 27。两边同时减去 12，得到 3x = 15；再除以 3，得到 x = 5。检验：3 × (5 + 4) = 3 × 9 = 27。
3(x + 4) = 27 → 3x + 12 = 27 → 3x = 15 → x = 5
When an equation contains fractions, multiply every term on both sides by the lowest common multiple of the denominators. This clears all the fractions in one step and turns the question into an ordinary linear equation.
当方程含有分数时，把等式两边每一项都乘以各分母的最小公倍数。这样一步就能清除所有分数，把题目变成普通的线性方程。
− 1 = 12, then 2x = 13, so x = 6.5. Check: (2 × 6.5 − 1) ÷ 4 = (13 − 1) ÷ 4 = 12 ÷ 4 = 3.
例题：(2x − 1) ÷ 4 = 3。两边同时乘以 4，得到 2x − 1 = 12；于是 2x = 13，x = 6.5。检验：(2 × 6.5 − 1) ÷ 4 = (13 − 1) ÷ 4 = 12 ÷ 4 = 3。
2, so multiplying through by 12 gives 4x + 3x = 84, that is 7x = 84, and therefore x = 12.
第二种类型的题目中，未知数出现在两个不同分数的分子上，例如 x ÷ 3 + x ÷ 4 = 7。3 与 4 的最小公倍数是 12，两边同乘 12 得到 4x + 3x = 84，即 7x = 84，所以 x = 12。
Watch out for a bracket multiplied by a negative number: −2(x − 5) becomes −2x + 10, not −2x − 10. Writing the expansion on its own line before solving prevents this classic error.
要特别小心括号前是负数的情况：−2(x − 5) 展开为 −2x + 10，而不是 −2x − 10。把展开式单独写一行，可以有效避免这个经典错误。
4. Unknowns on Both Sides | 未知数出现在等式两边
If the unknown appears on both sides, collect all the x-terms on the side where the coefficient will be positive, and collect all the numbers on the other side. The equation still behaves like a balance: whatever you do to one side, you must do to the other.
如果未知数出现在等式两边，就把所有含 x 的项移到系数为正的那一边，把所有常数项移到另一边。等式仍然像一台天平：对一边做什么，就必须对另一边做同样的事。
6. Check: left side 4 × 6 − 3 = 21 and right side 2 × 6 + 9 = 21, so the two sides agree.
例题：解 4x − 3 = 2x + 9。两边同时减去 2x，得到 2x − 3 = 9；再两边同时加 3，得到 2x = 12，所以 x = 6。检验：左边 4 × 6 − 3 = 21，右边 2 × 6 + 9 = 21，两边相等。
4x − 3 = 2x + 9 → 2x − 3 = 9 → 2x = 12 → x = 6
Subtracting the smaller coefficient of x is usually the better choice, because it keeps the remaining coefficient positive and avoids extra negative signs. If the coefficients are equal, the x-terms cancel and you are left with either a true statement (infinitely many solutions) or a false one (no solution).
通常更好的做法是减去较小的 x 系数，这样剩下的系数为正，可以避免多余的负号。如果两边 x 的系数相等，x 项会消掉，剩下的要么是真命题（无穷多解），要么是假命题（无解）。
- 3x + 8 = 7x − 4 → 12 = 4x → x = 3
- 3x + 8 = 7x − 4 → 12 = 4x → x = 3
- 5(x − 2) = 3x + 6 → 5x − 10 = 3x + 6 → 2x = 16 → x = 8
- 5(x − 2) = 3x + 6 → 5x − 10 = 3x + 6 → 2x = 16 → x = 8
When a question says "show that x = ..." ute the given value into both sides and demonstrate that they are equal, writing each side's value explicitly.
当题目要求"证明 x = ……"时，不能只是解方程；你必须把给定的值分别代入两边，并明确写出每一边的数值，证明它们相等。
5. Rearranging Formulae | 公式变形
Rearranging a formula uses exactly the same inverse-operation method, except that the answer is an expression rather than a number. Treat every letter other than the subject as if it were a number, and keep the working in a single vertical chain of equivalent formulae.
公式变形使用完全相同的逆运算方法，只是答案是一个表达式而不是一个数。把除"主变量"以外的每个字母都当作数字来处理，并把过程写成一条纵向的等价公式链。
= u + at. Subtract u from both sides: v − u = at. Divide both sides by t: a = (v − u) ÷ t.
例1：把 v = u + at 变形为以 a 为主变量。两边同时减去 u，得到 v − u = at；两边同时除以 t，得到 a = (v − u) ÷ t。
subject of P = 2(l + w). Divide both sides by 2: P ÷ 2 = l + w. Subtract l: w = P ÷ 2 − l.
例2：把 P = 2(l + w) 变形为以 w 为主变量。两边同时除以 2，得到 P ÷ 2 = l + w；再减去 l，得到 w = P ÷ 2 − l。
ly both sides by 9: 9C = 5(F − 32). Divide by 5: 9C ÷ 5 = F − 32. Add 32: F = 9C ÷ 5 + 32.
例3：把 C = 5(F − 32) ÷ 9 变形为以 F 为主变量。两边同乘 9，得到 9C = 5(F − 32)；除以 5，得到 9C ÷ 5 = F − 32；再加 32，得到 F = 9C ÷ 5 + 32。
ubject of y = 3x + kx gives y = x(3 + k) and therefore x = y ÷ (3 + k), provided 3 + k ≠ 0.
如果主变量在公式中出现两次，先把含它的所有项集中到一边，再把它提取公因式。例如把 y = 3x + kx 变形为以 x 为主变量，可得 y = x(3 + k)，因此 x = y ÷ (3 + k)，其中 3 + k ≠ 0。
6. Solving Linear Inequalities | 解一元一次不等式
Linear inequalities are solved in almost the same way as linear equations, using inverse operations. The one crucial difference is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. Adding or subtracting a negative number does not reverse it.
一元一次不等式的解法与线性方程几乎相同，同样使用逆运算。唯一关键的区别是：当两边同时乘以或除以一个负数时，不等号的方向必须反转；而加上或减去一个负数时，不等号方向不变。
Worked example 1: solve 3x − 5 < 10. Add 5 to both sides: 3x < 15. Divide by 3 (positive, so the sign stays): x < 5.
例题1：解 3x − 5 < 10。两边同时加 5，得到 3x < 15；再除以 3（正数，不等号不变），得到 x < 5。
Worked example 2: solve −2x ≥ 8. Divide both sides by −2 and reverse the sign: x ≤ −4.
例题2：解 −2x ≥ 8。两边同时除以 −2，并反转不等号，得到 x ≤ −4。
→ x < 5&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;−2x ≥ 8 → x ≤ −4
fies the inequality. If the inequality is written with the unknown on the right, such as 7 > x, simply read it backwards as x < 7 before you draw anything.
在数轴上表示时，< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆点，然后给满足不等式的区域涂色。如果不等式中未知数写在右边，例如 7 > x，先把它反过来读成 x < 7 再动手作图。
, then the integer solutions are −1, 0, 1, 2 and 3; note that −2 is excluded but 3 is included, and that this distinction carries marks.
有些题目要求写出整数解。如果 x 是整数且 −2 < x ≤ 3，那么整数解是 −1、0、1、2 和 3；注意 −2 不包含在内而 3 包含在内，这个区别是得分点。
7. Plotting Straight-Line Graphs | 绘制直线图像
To plot a straight line from its equation, build a small table of values, plot the points on a coordinate grid, then join them with a single straight line drawn with a ruler and extended across the whole grid. Two points are enough mathematically, but three points make plotting errors obvious.
要根据方程绘制直线，先列一张小的数值表，在坐标网格上描点，然后用直尺把点连成一条直线，并延伸到整个网格。从数学上说两点就够，但取三点能立刻暴露描点错误。
| x | −1 | 0 | 1 | 2 | 3 |
| y = 2x − 1 | −3 | −1 | 1 | 3 | 5 |
Each pair of values in the table gives one coordinate: (−1, −
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