Linear Equations and Graphs | 线性方程与图像

📚 Linear Equations and Graphs | 线性方程与图像

This guide supports the Year 9 Unit 5 homework by covering linear equations and straight-line graphs. A linear equation produces a straight line when plotted, and its algebra gives you powerful tools for solving problems in mathematics and science.

本指南针对九年级第五单元作业，涵盖线性方程与直线图像。线性方程在坐标系中画出的图像是一条直线，其代数方法为你提供了解决数学和科学问题的有力工具。

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

A linear equation is an equation in which the highest power of the variable is 1. It can usually be written in the form ax + b = 0 or y = mx + c, where a, b, m and c are constants.

线性方程是指未知数的最高次数为 1 的方程。它通常可以写成 ax + b = 0 或 y = mx + c 的形式，其中 a、b、m、c 是常数。

Examples include 2x + 3 = 11, 5y − 7 = 3y + 9, and y = 4x − 1. The graph of y = mx + c is always a straight line.

例如 2x + 3 = 11、5y − 7 = 3y + 9 以及 y = 4x − 1。y = mx + c 的图像总是一条直线。

Standard form: y = mx + c

标准形式：y = mx + c

2. Solving Two-Step Equations | 解两步方程

To solve a two-step equation, undo the operations in reverse order. First remove the constant term by adding or subtracting, then divide or multiply to isolate the variable.

解两步方程时，要按相反顺序进行逆运算。先通过加法或减法消去常数项，再通过除法或乘法将未知数的系数化为 1。

Example: Solve 2x + 3 = 11. Subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4.

例题：解方程 2x + 3 = 11。两边同时减 3：2x = 8。然后两边同时除以 2：x = 4。

Check your answer: Substitute x = 4 into the original equation to see that 2(4) + 3 = 11.
检验答案：将 x = 4 代回原方程，可得 2(4) + 3 = 11。

3. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law. Then simplify both sides before applying inverse operations.

当方程中含有括号时，先用分配律展开括号。然后化简两边，再进行逆运算。

Example: Solve 3(x − 2) = 12. Expand to get 3x − 6 = 12. Add 6 to both sides: 3x = 18. Divide by 3: x = 6.

例题：解方程 3(x − 2) = 12。展开得 3x − 6 = 12。两边同时加 6：3x = 18。再除以 3：x = 6。

A common mistake is to forget to multiply every term inside the bracket by the outside factor. Always write the expanded form carefully.

一个常见错误是忘记将括号内的每一项都与括号外的因数相乘。请务必仔细写出展开后的形式。

4. Equations with Variables on Both Sides | 变量在方程两边的方程

If a variable appears on both sides of the equation, collect the variable terms on one side and the constant terms on the other side.

如果未知数出现在方程的两边，应把含未知数的项移到一边，把常数项移到另一边。

Example: Solve 5x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Subtract 2: 2x = 8. Divide by 2: x = 4.

例题：解方程 5x + 2 = 3x + 10。两边同时减 3x：2x + 2 = 10。再两边同时减 2：2x = 8。最后除以 2：x = 4。

Always keep the balance of the equation by doing the same operation to both sides.

必须始终对方程两边同时进行相同的运算，以保持等式的平衡。

5. Rearranging into y = mx + c | 改写为 y = mx + c

Linear graphs are easiest to analyse when the equation is in the form y = mx + c. To rearrange, make y the subject by using inverse operations.

当方程写成 y = mx + c 的形式时，线性图像最容易分析。改写的方法是使用逆运算，把 y 单独放在等号左边。

Example: Rearrange 2y − 6 = 4x. Add 6: 2y = 4x + 6. Divide by 2: y = 2x + 3.

例题：将 2y − 6 = 4x 改写。两边加 6：2y = 4x + 6。两边除以 2：y = 2x + 3。

In this form, the gradient is m = 2 and the y-intercept is c = 3.

在这种形式下，斜率为 m = 2，y 轴截距为 c = 3。

6. Plotting Linear Graphs | 绘制线性图像

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Linear Equations and Graphs | 线性方程与图像