⟷ 代入法解方程组计算器（二元/三元一次，含步骤）Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps)

使用说明与常见问题How-to & FAQ

「代入法解方程组计算器（二元/三元一次，含步骤）」是什么？What is the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) tool?

用代入法解二元一次方程组 a₁x + b₁y = c₁、a₂x + b₂y = c₂，并写出每一步：从一个方程解出一个变量、代入另一个方程、求值、回代。判断无解（平行）或无穷多解（重合），并可切换到三元一次方程组。本地处理。Solves a₁x + b₁y = c₁ and a₂x + b₂y = c₂ by substitution, showing each step: isolate a variable in one equation, substitute it into the other, solve, and back-substitute. Detects no solution (parallel lines) or infinitely many (the same line), and also handles 3×3 systems. Processed locally.

如何使用「代入法解方程组计算器（二元/三元一次，含步骤）」？How do I use the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) tool?

在「代入法解方程组计算器（二元/三元一次，含步骤）」中，填写「方程组」、「a1」、「b1」、「c1」、「a2」等参数，结果会随输入实时计算并显示在下方。In the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps): Fill in "System", "a1", "b1", "c1", "a2"; results are recalculated instantly as you type.

代入法怎么做？How does the substitution method work?

从一个方程中解出一个变量（如 y = 5 − 2x），代入另一个方程得到只含一个变量的方程，解出后再回代求另一个变量。Solve one equation for a variable (such as y = 5 − 2x), substitute it into the other to get a one-variable equation, solve it, then back-substitute for the other variable.

什么时候无解或有无穷多解？When is there no solution or infinitely many?

当 a₁b₂ − a₂b₁ = 0 时两直线斜率相同：常数也成比例就是同一条直线（无穷多解），否则平行（无解）。When a₁b₂ − a₂b₁ = 0 the lines have the same slope: if the constants are in the same ratio they are the same line (infinitely many solutions), otherwise they are parallel (none).

三元方程组怎么输入？How do I enter a 3×3 system?

每行写一个方程的 4 个数 a b c d，表示 ax + by + cz = d。例如 x + y + z = 6 写作 1 1 1 6。Write one equation per line as four numbers a b c d for ax + by + cz = d. For example x + y + z = 6 is 1 1 1 6.

能举个计算实例吗？Can you show a worked example?

例如在「代入法解方程组计算器（二元/三元一次，含步骤）」中填 a1 = 2，b1 = 1，c1 = 5，a2 = 1，b2 = -1，c2 = 1，结果区显示：x：2；y：1For example, entering a1 = 2, b1 = 1, c1 = 5, a2 = 1, b2 = -1, c2 = 1 in the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) gives: x: 2; y: 1

哪些输入会被检查？Which inputs are validated?

「代入法解方程组计算器（二元/三元一次，含步骤）」会检查这些输入——「a1」：所选方式用到时必须填写；「b1」、「c1」、「a2」、「b2」、「c2」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) checks its inputs — "a1": required when the chosen option uses it; "b1", "c1", "a2", "b2", "c2": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「代入法解方程组计算器（二元/三元一次，含步骤）」的设置：「方程组」可选 二元 2×2、三元 3×3（默认 二元 2×2）；「a1」默认 2；「b1」默认 1；「c1」默认 5；「a2」默认 1；「b2」默认 -1；「c2」默认 1。每次生成结果都按当前设置处理。Settings in the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) — "System": Two variables 2×2, Three variables 3×3 (default Two variables 2×2); "a1" default 2; "b1" default 1; "c1" default 5; "a2" default 1; "b2" default -1; "c2" default 1. The result always uses the current settings.

计算精度如何？How precise are the results?

「代入法解方程组计算器（二元/三元一次，含步骤）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「代入法解方程组计算器（二元/三元一次，含步骤）」免费吗？需要注册吗？Is the Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) free? Do I need an account?

「代入法解方程组计算器（二元/三元一次，含步骤）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Substitution Method Calculator (2×2 and 3×3 Linear Systems, With Steps) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.