带变量的方程
代数是处理数论、几何和分析的最古老的数学分支之一,被称为代数。根据一些定义。它被定义为对数学符号和规律的分析,以及对这些数学符号的操纵。
Algebra also includes a wide range of topics, from solving basic problems to studying abstractions. basically focused on variable which has undefined values .The branch of mathematics that deals with variables and the arithmetic operations across these symbols. These variables do not have any fixed values . In our day to day life problem. There are certain values that keep on changing.
在代数中,这些值或变量通常由 x、y、z、p 或 q 表示。这些变量通过加法、减法、乘法和除法的各种算术运算进行操作,目的是找到值。
表达式
通过使用数学运算(例如减法、加法、乘法和除法)组合的项的组合称为表达式。
- 常数是一个固定的数值。
- 变量是没有固定值的符号。
- 常数、单个变量或变量和常数与乘法或除法相结合的组合被定义为项。
代数表达式示例:3x + 9 , 5x + 10 …
方程中的变量
变量定义为表示数值或数字的字母字符。在代数中,变量用于表示未知量或未知值。
这些字母最常见的是,“a”、“b”、“c”、“x”、“y”和“z”用作方程中的变量。变量是定义另一个变量或方程的值的术语
For example, in equation, 2x – 8 = 20
x → Variable
2 → coefficient of x
8 and 20→ constants
‘-’ → operator
变量类型
有两种类型的变量,即
- 因变量:其值取决于在其条件下另一个变量的估计的变量称为因变量......
For example: Consider the condition y = 5x + 3. In this equation, the value of the variable ‘y’ changes as per the adjustments in the value of ‘x’.
- 自变量:自变量是描述其值与变化无关的变量的变量。假设如果 x 和 y 是代数方程中的两个变量,则 x 的每个值都与 y 的任何其他值相关联。这里'y'的值被称为x值的函数,称为自变量,'y'的值称为因变量。
For example: In this function y = x2, here x is an independent variable and y is a dependent variable whose value is dependent on the value of x.
变量方程或代数表达式的一些示例:
- y = 5x +10
- 3y = 99x + 9
- 3x + 9
- 3x2 + 5x
In this way, a variable is used in an equation to find the other variable or to complete the equation…
示例问题
问题 1:求 x 的值:5(x + 2) = 3x – 8
解决方案:
We have
5(x + 2) = 3x – 8 {here x is independent variable}
5x + 10 = 3x – 8
by adding or subtracting like terms together
5x – 3x = – 8 + 10
2x = 2
x = 1
here the value of x is 1
问题 2:简化方程 3x + 2y = 16:如果 x = 2,求 y?
解决方案:
Given equation : 3x + 2y = 16
here x = 2
In above equation y is dependent variable whose value is dependent on the value of x
put the value of x in equation
3 (2) + 2y = 16
6 + 2y = 16
2y = 16 – 6
2y = 10
y = 10/2
y = 5
Therefore the value of y is 5
问题 3:简化方程:2x + 5y 2 = 40 并找到 x 的值,如果 y = 2?
解决方案:
Given equation : 2x + 5y2 = 40
y = 2
in this equation x is dependent variable whose value is depend on the value of y . whereas y is independent
we have
2x + 5y2 = 40
2x + 5 (2)2 = 40
2x + 5 (4) = 40
2x + 20 = 40
2x = 40 – 20
2x = 20
x = 10
hence the value of x is 10
问题4:一个袋子里有30支笔。写出 x 个袋子中的笔数的变量表达式(代数表达式)。
解决方案:
The number of pens in one bag = 25
The number of bags = x
So the number of pens in x bags = 25x
here 25x is the variable expression
lets assume numbers of bags x = 2
then the solution will be 25x = 25 × 2 = 50 pens
问题 5:求解方程 5x – 10 = 3x – 8。
解决方案:
Given, 5x – 10 = 3x – 8
Adding 10 on both sides,
5x – 10 + 10 = 3x – 8 + 10
5x = 3x + 2
Subtract 3x from both sides,
5x – 3x = 3x + 2 – 3x
2x = 2
Dividing both sides of the equation by 2,
2x/2 = 2/2
x = 1
问题 6:计算给定的变量表达式 a = 7; b = -3 和 c = 2, 3ab+7bc+9ca。
解决方案:
The given algebraic expression is 3ab+7bc+9ca
Substitute the below values in the above expression:
a=7;b=−3;c=2
3ab+7bc+9ca
=3(7)(−3)+7(−3)(2)+9(2)(7)
=−63−42+126
=−105 + 126
= 21
问题 7:Arun 有 6 个弹珠。 Annie 拥有的弹珠数量是 Arun 拥有的弹珠数量的 5 倍多。写出表达式并找出 Arun 和 Annie 一起拥有的弹珠总数?
解决方案:
Here given Arun marbles (x) = 6
lets assume the marbles annie has (y)
As per the question : Annie has 5 more than 3 times the number of marbles Arun has
Therefore, y = 5 + 3x is an expression
= 5 + 3(6) { Arun marbles (x) = 6 }
= 5 + 18
y = 23
hence the number of marbles annie has y = 23
and total number of marbles = x + y
= 6 + 23
= 29 marbles