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进阶
What value of p is the solution to the equation \( 4p-5p=7 \)?
原库参考解析
一元一次方程与代数求值进阶
\( y-39=px \)
The given equation relates the positive numbers p , x , and y . Which equation correctly expresses y in terms of p and x ?
原库参考解析
一元一次方程与代数求值挑战
\( 0.10x+0.20y=0.19(x+y) \)
The equation gives a volume x, in gallons, of a 10% saltwater solution that could be mixed with a volume y, in gallons, of a 20% saltwater solution to produce a 19% saltwater solution. According to this equation, what volume, in gallons, of the 20% saltwater solution could be mixed with 80.0 gallons of the 10% saltwater solution to produce a 19% 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 \( \frac{3x}{2}-12=\frac{21}{2}-12 \) , what is the value of \( 3x \)?
原库参考解析
一元一次方程与代数求值进阶
A geologist is analyzing a sample of a coarse-grained igneous rock. The equation \( 3.00x+4.00y=98.0 \) gives the relationship between the estimated volumes of mineral X and mineral Y found in the sample, where x is the estimated volume, in cubic centimeters cm³, of mineral X and y is the estimated volume, in cm³, of mineral Y. If the estimated volume of mineral Y in the sample is 12.8 cm³, what is the estimated volume, in cm³, of mineral X in the sample?
原库参考解析
一元一次方程与代数求值进阶
If \( \frac{1-x}{5}-81=\frac{7\left(1-x\right)}{8} \), what is the value of \( 2−2x \)?
原库参考解析
一元一次方程与代数求值挑战
\( \frac{x-8}{8}+7=\frac{x+7}{3} \)
The solution to the given equation 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进阶
\( x+75=175 \)
What value of x is the solution to the given equation?
原库参考解析
一元一次方程与代数求值进阶
If \( 5x+4=34 \), what is the value of \( 50x+40 \)?
参考答案
一元一次方程与代数求值进阶
If \( 15x=13x+10 \), what is the value of \( 7x \)?
参考答案
一元一次方程与代数求值进阶
\( 5x+12=2(8+6x) \)
Which of the following is equivalent to the given equation?
参考答案
一元一次方程与代数求值进阶
\( x=\frac{w-23y+z}{46wy} \)
The given equation relates the variables w, x, y and z where the values of w, x, and z are distinct and positive. Which expression is equivalent to y?
参考答案
