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
In the given system of equations, a and b are constants. The graphs of these equations in the xy-plane intersect at the point (4, y). What is the value of a?
原库参考解析
二元一次方程组求解挑战
\( 8x+y=5 \)
\( y=9x+1 \)
The solution to the given system of equations is (x, y). What is the value of x?
原库参考解析
二元一次方程组求解进阶
x +3=-2y +5
x -3 =2y +7
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基础
x =5
y = x -8
Which of the following points (x,y ) is the solution to the given system of equations in the xy-plane?
原库参考解析
二元一次方程组求解进阶
A sample of a certain alloy has a total mass of 50.0 grams and is 50.0% silicon by mass. The sample was created by combining two pieces of different alloys. The first piece was 30.0% silicon by mass and the second piece was 80.0% silicon by mass. What was the mass, in grams, of the silicon in the second piece?
原库参考解析
二元一次方程组求解基础
\( x=3 \)
\( y=\left(15-x\right)^2 \)
A solution to the given system of equations is (x, y). What is the value of xy?
原库参考解析
二元一次方程组求解基础
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x, y) to this system?
原库参考解析
二元一次方程组求解基础
\( x=4 \)
\( y=5-x \)
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挑战
\( y=\frac{1}{3}x-14 \)
\( y=-x+18 \)
The solution to the given system of equations is (x, y). What is the value of x?
原库参考解析
二元一次方程组求解挑战
A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater sold 30 tickets for a total of $300, how much, in dollars, was spent on adult tickets? (Disregard the $ sign when gridding your answer.)
原库参考解析
二元一次方程组求解挑战
5x +14y =45
10x + 7y =27
The solution to the given system of equations is (x , y ) . What is the value of xy ?
原库参考解析
二元一次方程组求解挑战
Store A sells raspberries for $5.50 per pint and blackberries for $3.0 per pint. Store B sells raspberries for $6.5 per pint and blackberries for $8.00 per pint. A certain purchase of raspberries and blackberries would cost $37.00 at Store A or $66.00 at Store B. How many pints of blackberries are in this purchase?
原库参考解析
