这部分需要会什么？
- 熟练去括号、通分、移项和合并同类项
- 区分求未知数与求一个代数式的值
- 识别无解、唯一解和恒等式
核心概念与公式
一元一次方程整理后可以写成 \(ax+b=c\)。当 \(a\ne0\) 时，\(x=(c-b)/a\)。等式两边做同一种合法运算，解保持不变；除法的除数不能为零。
若未知数项消去，留下真等式（如 \(0=0\)），则原允许范围内所有数都是解；留下假等式（如 \(0=5\)），则无解。题目若只问 \(2x+3\)，可以把它当成整体，不必一定先求 \(x\)。
做题时怎样使用
- 先看题目最终要求什么，并标出分母或其他取值限制
- 去括号、通分，整理未知数项与常数项；有参数时先判断能否除以它
- 得到答案后代回原式，尤其检查负号和括号
容易混淆的地方
\(-(x-3)=-x+3\)，不是 \(-x-3\)。若等式中某个系数可能为零，直接除以它会漏掉特殊情况。
用一个例题串起方法
知识点教学例题已知 \(5(2x-1)=3x+16\)，求 \(2x+3\)。
展开解题思路 ＋
展开得 \(10x-5=3x+16\)，于是 \(7x=21\)，\(x=3\)。题目要求的是 \(2x+3=9\)，不能把中间结果 3 当作最终答案。
本知识点题库
先独立作答，再查看答案与解析。
一元一次方程与代数求值挑战
If \( 16+\frac{1}{4}(7x-6)=\frac{3}{4}(7x-6) \) , what is the value of \( 7x−6 \)?
原库参考解析
一元一次方程与代数求值进阶
\( 0.10x+0.20y=0.18(x+y) \)
This equation represents mixing a volume x, in gallons of a 10% saltwater solution with a volume y , in gallons of a 20% saltwater solution to produce an 18% saltwater solution. According to this equation, what volume, in gallons, of the 20% saltwater solution could be mixed with 45.0 gallons of the 10% saltwater solution to produce an 18% saltwater solution? (Assume that the volume of the mixture is the sum of the volumes of the two solutions before they were mixed.)
原库参考解析
一元一次方程与代数求值挑战
If \( \frac{9x}{2}-11=\frac{45}{2}-11 \), what is the value of \( 9x \)?
原库参考解析
一元一次方程与代数求值进阶
If \( 3+\frac{1-x}{1}=\frac{2(1-x)}{1} \) , what is the value of \( 1−x \)?
原库参考解析
一元一次方程与代数求值进阶
A piece of a metal alloy called delta metal has a total mass of 200.0 grams. The equation \( c+z+3.1=200.0 \) relates the mass of copper c, in grams, and the mass of zinc z, in grams, in this piece of metal. If this piece of metal contains 110.0 grams of copper, how many grams of zinc does it contain?
原库参考解析
一元一次方程与代数求值进阶
\( 40x+36y=720 \)
Tom saved $40 each month for x months, and Karla saved $36 each month for y months. Together, Tom and Karla saved a total of $720. If Tom saved for 9 months, for how many months did Karla save?
原库参考解析
一元一次方程与代数求值进阶
\( \frac{v}{r}-12=2s \)
The given equation relates the positive numbers r, s, and v. Which equation correctly expresses v in terms of r and s?
原库参考解析
一元一次方程与代数求值进阶
\( 11j+9k=m \)
The given equation relates the numbers j, k, and m. Which equation correctly expresses k in terms of j and m?
原库参考解析
一元一次方程与代数求值进阶
If \( \frac{x}{y}=6 \) and\( \frac{7x}{ny} \) \( =6 \), what is the value of n?
原库参考解析
一元一次方程与代数求值基础
If \( 6x=9 \), what is the value of \( 48x \)?
原库参考解析
一元一次方程与代数求值挑战
If \( \frac{x+6}{5}=\frac{x+6}{13} \), the value of \( x+6 \) is between which of the following pairs of values?
原库参考解析
一元一次方程与代数求值挑战
\( R=\frac{0.36ℓ}{A} \)
The given equation relates the resistance R , in ohms, of a wire to its length \( ℓ \), in meters, and its cross-sectional area A , in square meters. Which equation correctly expresses the length, in meters, of the wire in terms of its resistance, in ohms, and its cross-sectional area, in square meters?
原库参考解析
