这部分需要会什么？
- 用代入法或消元法解二元一次方程组
- 把交点坐标解释为同时满足两个条件的数对
- 直接求所需的组合量
核心概念与公式
方程组的解 \((x,y)\) 必须同时满足每一个方程。代入法适合某个变量已经被单独表示的情形；消元法通过倍乘、相加或相减消去一个变量。图象上，两条直线的交点就是方程组的解。
若只要求 \(x+y\)、\(2x-y\) 等组合，可以先观察能否把两个方程线性组合成目标式。这样常常比把两个未知数都求出来更快。
做题时怎样使用
- 对齐两个方程的同类项，先选最容易消掉的变量
- 消元或代入得到一个变量，再代回求另一个
- 检查两个原方程，并确认题目问的是单个变量还是组合量
容易混淆的地方
将方程乘一个数时，每一项和等号右边都要乘；两个方程相减时，要给第二个方程的整行加括号。
用一个例题串起方法
知识点教学例题已知 \(2x+y=11\)，\(x-y=1\)，求 \(x+y\)。
展开解题思路 ＋
两式相加得 \(3x=12\)，故 \(x=4\)。代入 \(x-y=1\) 得 \(y=3\)，于是 \(x+y=7\)。数对 \((4,3)\) 同时满足两个方程。
本知识点题库
先独立作答，再查看答案与解析。
二元一次方程组求解挑战
\( y=-9x-6 \)
\( y=9x+10 \)
The solution to the given system of equations is (x, y) . What is the value of y?
原库参考解析
二元一次方程组求解进阶
\( (x−6)−8 \)\( \left(y+2\right) \)\( =131 \)
\( \left(x-6\right)+8\left(y+2\right)=438 \)
The solution to the given system of equations is (x, y). What is the value of \( 10\left(x-6\right) \) ?
原库参考解析
二元一次方程组求解挑战
\( x=2+8y \)
\( x=4-8y \)
The solution to the given system of equations is (x, y). What is the value of 2x?
原库参考解析
二元一次方程组求解挑战
\( y=\frac{x}{3}+8 \)
\( y=-\frac{x}{3}+50 \)
The solution to the given system of equations is (x, y). What is the value of 2y?
原库参考解析
二元一次方程组求解进阶
\( x=2 \)
\( y=\frac{1}{x+8} \)
The graphs of the given equations intersect at the point (x , y) in the xy-plane. What is the value of y?
原库参考解析
二元一次方程组求解挑战
\( y=\frac{x}{3}+2 \)
\( y=-\frac{x}{3}+60 \)
The solution to the given system of equations is ( x, y ) . What is the value of 2y?
原库参考解析
二元一次方程组求解进阶
\( \frac{x-7}{5}+\frac{y+8}{4}=0 \)
\( 3x+4y=0 \)
The solution to the given system of equations is (x, y). What is the value of x + y?
原库参考解析
二元一次方程组求解进阶
Jackson mows lawns in the summer. When Jackson mows a lawn with a fence he charges $1.03 per minute. When Jackson mows a lawn without a fence, he charges $0.56 per minute. Jackson spends a total of 15 hours mowing lawns with and without a fence in a week and charges a total of $701.40 for mowing lawns during that week. How many minutes does Jackson spend mowing lawns with a fence during that week?
原库参考解析
二元一次方程组求解进阶
\( \frac{9}{2}x+7y=30 \)
\( \frac{5}{2}x+7y=26 \)
The solution to the given system of equations is (x, y). What is the value of \( \frac{19}{2}x+21y \)?
原库参考解析
二元一次方程组求解挑战
\( y=6x+17 \)
\( -7x-y=35 \)
What is the solution (x, y) to the given system of equations?
原库参考解析
二元一次方程组求解挑战
\( 5\left(3x\right)+4\left(8y\right)=16 \)
\( -5\left(3x\right)+4\left(8y\right)=16 \)
The solution to the given system of equations is (x, y). What is the value of \( 3x−8y \)?
原库参考解析
二元一次方程组求解进阶
\( y=x^{2}+1.7 \)
\( y=1.7−x \)
Which graph represents the given system of equations?
原库参考解析
