七年级数学上册第三章一元一次方程知识点与练习

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         

一元一次方程

方程：含           的等式叫做方程.

方程的解：使方程的等号左右两边相等的                ，就是方程的解。

解 方 程：求         的过程叫做解方程。

一元一次方程

只含有一个未知数（元），未知数的最高次数是1的整式方程叫做一元一次方程。

等式的基本性质

等式的性质1：等式的两边同时加（或减）        （      ），结果仍相等。

即：如果a=b，那么a±c=b    。

等式的性质2：等式的两边同时乘         ，或除以               数，结果仍相等。

即：如果a=b，那么ac =bc  ；  或  如果a=b（     ），那么a/c =b/c

△分数的基本的性质

    分数的分子、分母同时乘以或除以同一个不为0的数，分数的值不变。

即：==（其中m≠0）

1、在①；②；③；④中，等式有_____________，方程有_____________．

2、根据“的倍与的和比的小”，可列方程为____            ___．

3、若(a－1)x|a|＋3＝－6是关于x的一元一次方程，则a＝＿＿

步骤	名 称	方   法	依  据	注 意 事 项
1	去分母	在方程两边同时乘以所有分母的最小公倍数（即把每个含分母的部分和不含分母的部分都乘以所有分母的最小公倍数）	等式性质2	1、不含分母的项也要乘以最小公倍数；2、分子是多项式的一定要先用括号括起来。
2	去括号	去括号法则（可先分配再去括号）	乘法分配律	注意正确的去掉括号前带负数的括号
3	移项	把未知项移到方程的一边（左边），常数项移到另一边（右边）	等式性质1	移项一定要改变符号
4	合并 同类项	分别将未知项的系数相加、常数项相加	1、整式的加减； 2、有理数的加法法则	单独的一个未知数的系数为“±1”
5	系数化为“1”	在方程两边同时除以未知数的系数（方程两边同时乘以未知数系数的倒数）	等式性质2	不要颠倒了被除数和除数（未知数的系数作除数——分母）
*6	检根 x=a	方法：把x=a分别代入原方程的两边，分别计算出结果。    ①　若 左边＝右边，则x=a是方程的解； ②　若 左边≠右边，则x=a不是方程的解。 注：当题目要求时，此步骤必须表达出来。

4、如果，那么a=       ，其根据是                             ．

一元一次方程课后作业

解一元一次方程：

1、方程的解是_______．

2、当x=      时，代数式与代数式的值相等．

3、若与有相同的解，那么___  _        ___．

4、代数式与互为相反数，则　　　　　．

5、解方程：

       8(3x－1)－9(5x－11)－2(2x－7)=30

             ④

⑤     

利用已学知识，构造一元一次方程

1、根据绝对值或平方数相加等于零（注意：，）

（1）已知，求和的值.

（2）若，求的值．

2、方程中有未知字母，根据方程的解，求未知字母

（1）已知是方程的解，求的值.

（2）已知时，代数式的值是14，求时代数式的值．

3、根据代数式值相等、同类项或相反数的知识

（1）若代数式与代数式的值相等，求的值.

（2）当、取什么值时，单项式与是同类项？

一元一次方程应用题

1、数字问题

（1）已知三个连续偶数的和是2004，求这三个偶数各是多少？

（2）一个两位数，十位上的数字比个位上的数字小5，若此两位数的两个数字位置交换，得一新两位数，那么新两位数与原两位数大45，求新两位数与原两位数的积是多少？

2、调配问题

（1）有两个工程队，甲工程队有32人，乙工程队有28人，如果是甲工程队的人数是工程队人数的2倍，需从乙工程队抽调多少人到甲工程队？

（2）某班同学利用假期参加夏令营活动，分成几个小组，若每组7人还余1人，若每组8人还缺6人，问该班分成几个小组，共有多少名同学？

3、年龄问题

（1）某同学今年15岁，他爸爸今年39岁，问几年以后，爸爸的年龄是这位同学年龄的2倍？

（2）三位同学甲乙丙，甲比乙大1岁，乙比丙大2岁，三人的年龄之和为41，求乙同学的年龄.

4、销售问题

（1）某产品按原价提高40%后打八折销售，每件商品赚270元，问该商品原标价多少元？现销售价是多少？

（2）甲乙两件衣服的成本共500元，商店老板为获取利润，决定将家服装按50%的利润定价，乙服装按40%的利润定价，在实际销售时，应顾客要求，两件服装均按9折出售，这样商店共获利157元，求甲乙两件服装成本各是多少元？

5、工程问题

（1）一个水池安有甲乙丙三个水管，甲单独开12h注满水池，乙单独开8h注满,丙单独开24h可排掉满池的水，如果三管同开，多少小时后刚好把水池注满水？

（2）某工程，甲单独完成续20天，乙单独完成续12天，甲乙合干6天后，再由乙继续完成，乙再做几天可以完成全部工程?

6、路程问题

（1）甲乙两个人在400米的环形跑道上同时同点出发，甲的速度是6米/秒，乙的速度是4米/秒，乙跑几圈后，甲可超过乙一圈？

（2）甲乙两站相距300km，一列慢车从甲站开往乙站，每小时行40km，一列快车从乙站开往甲站，每小时行80km，已知慢车先行1.5h，快车再开出，问快车开出多少小时后与慢车相遇？