Friends we all know about Numbers and how to combine them using various operations like addition, subtraction, multiplication and division. This area of study is what we call as Arithmetic. The more advanced area of study of Algebra are distinct from arithmetic in which addition or basic operations to specific numbers involves entities, what we called as variables that have no particular value or we specify it as unknown value. Variables in common are generally denoted by upper or lower case letters. Some of the basic examples to show variables are “3a” , “x”, “ 7 + d” here a, x and d are variables having unknown values.
Let’s talk about algebraic expression in mathematics. An algebraic expressions is basically a collection of letters and numbers combined together to form an expression. These letters and numbers are combined together by the four basic arithmetic operations. Some of the examples of algebraic expressions are 7a, 7a + b, 7a – 4b, a / (a + b), a2, (a + b)2
Here all the numbers used in algebraic expression are called constants and all the letters used in the above expressions like “a” and “b” are variables. If the expression comes with no variables then the algebraic expression is stated as arithmetic expressions. For example 4 + 7 / 6
Variables are generally used to explain the general situations or real time situations and they can also be used to solve problems that in anyway would be very difficult or even impossible to solve. Whenever we are going to solve algebraic expressions or any kind of word problems, we will see the use of these applications.
Now I am going to discuss about another topic which is very important that is an equation. In simplest mathematical manner we can say that an equation is an allegation in which two algebraic expressions used are equal. It is further stated in two different ways that are:
The first case: the given equation is true for all the values of the variables. In such kind of situation, equation is called an identity. The example to show an identity or we can say that values of variables which is true for the given equation. Most simple example is
x + y = y + x
It is also known as commutative law of addition. Another most common and well known identity is the first binomial formula or algebraic formulas like
(a + b)2 = a2 + 2ab + b2
The second case: In this situation the equation is true for some values of the variables. In such kind of problems or equations what we need to do is to identify those values of the variables for which the equation is true. This process is known as solving the equation. Moving forward we will study how to solve equations but for instance let’s take an example to understand the case in better manner
3a + 1 = 7
in this case obviously
a = 2 is the solution.
The given example is of a linear equation. We will further study about quadratic equations, but for know the example of a quadratic equation is a2 – a – 2 = 0. If we are going to find out values for variables then, this equation has the two results that are:
a = -1 or a = 2
Now I am going in deep with the topic so the next topic is, How to evaluate an algebraic expression? To evaluate or to explore an algebraic expression refers to substitute or place specific values (desired values) for its variables. Let’s take a simple example of an algebraic expression to understand it better:
algebraic = 2a + 1
Now we are going to evaluate the given algebraic expressions. Now what we need to do is to try various values of variable which satisfy the given algebraic expression. Lets take a = 3, which provides us: algebraic = 2 x 3 + 1 = 7. we can say that the value of algebraic at a = 3 is 7. let’s take an example which is little tougher or a bit complex then the above one
Now if we use above algebraic and put value of a = 2b + 1, where b is another variable which gives
Algebraic = 2 x (2b + 1) + 1 = 4b + 2 + 1 = 4b + 3
Now we can further solve this by substituting different values for variable b.
Before proceeding further towards inequalities let’s talk about equivalent expressions: Two expressions are equivalent if their values are equal for all possible evaluations of the two expressions. In other words presenting them with an equality sign between the expressions gives an identity.
Now I am going to talk about eighth grade linear equations topic:
In simple mathematical manner we can say that any equation that when graphed produces a straight line, then the equation is called as Linear Equation or we can say that any equation is a linear if it can be written in the linear form : ax = b. Here x is the variable, a and b are the constants. The common form of a linear equation in the two variables like x and y is
y = mx + b
where x and y are two variables and m and b are two constants.
The constant m determines the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b is also known as Y-Intercept.
There are three possible solutions for the linear equations:
Unique Solutions: if a not equal to b then only possible solution: x = b/a
No Solution: If a = 0 and b is not equal to 0 then it has no solution.
Infinite Solutions or many solutions: if a = 0 and b = 0. In this solution 0x = 0 and there are infinite solutions for all the values of x.
Let’s take an example : 2a + 3 = a + 5
For solving this we need to subtract 3 from both the sides 2a = a + 2 that will result in a = 2.
The most important thing to understand is that, for solving a linear equation we need to recognize that the equation is linear or non linear and if linear then convert it to the simplest form. Lets take an example to elaborate it well:
(a – 2)(a – 3) = (a + 1)(a + 2)
what we need to do is to apply distributive property to both the sides of the equation
axa – ax3 – ax2 + 2x3 = axa + 2xa + 1xa + 1x2
on further simplifying this we get
a2 – 5a + 6 = a2 + 3a + 2
This is what we get, we can further simplify it by subtracting a2 from both the sides.
Now after telling the basic concept behind linear equations, I am going to discuss about inequalities for class eighth. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. The most important thing to understand is the use of inequality symbols. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≤ less than or equal to etc.
In eighth standard we learn about linear inequality. So what linear inequality means is the first query comes in our mind. A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything on One side of a line on a graph. In mathematics a linear inequality is an inequality which involves a linear function.
The most important thing is to understand, how the inequality sign reverse when negative value comes. So to understand linear inequality, take an example: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12
-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the two.
We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.
Let’s talk about algebraic expression in mathematics. An algebraic expressions is basically a collection of letters and numbers combined together to form an expression. These letters and numbers are combined together by the four basic arithmetic operations. Some of the examples of algebraic expressions are 7a, 7a + b, 7a – 4b, a / (a + b), a2, (a + b)2
Here all the numbers used in algebraic expression are called constants and all the letters used in the above expressions like “a” and “b” are variables. If the expression comes with no variables then the algebraic expression is stated as arithmetic expressions. For example 4 + 7 / 6
Variables are generally used to explain the general situations or real time situations and they can also be used to solve problems that in anyway would be very difficult or even impossible to solve. Whenever we are going to solve algebraic expressions or any kind of word problems, we will see the use of these applications.
Now I am going to discuss about another topic which is very important that is an equation. In simplest mathematical manner we can say that an equation is an allegation in which two algebraic expressions used are equal. It is further stated in two different ways that are:
The first case: the given equation is true for all the values of the variables. In such kind of situation, equation is called an identity. The example to show an identity or we can say that values of variables which is true for the given equation. Most simple example is
x + y = y + x
It is also known as commutative law of addition. Another most common and well known identity is the first binomial formula or algebraic formulas like
(a + b)2 = a2 + 2ab + b2
The second case: In this situation the equation is true for some values of the variables. In such kind of problems or equations what we need to do is to identify those values of the variables for which the equation is true. This process is known as solving the equation. Moving forward we will study how to solve equations but for instance let’s take an example to understand the case in better manner
3a + 1 = 7
in this case obviously
a = 2 is the solution.
The given example is of a linear equation. We will further study about quadratic equations, but for know the example of a quadratic equation is a2 – a – 2 = 0. If we are going to find out values for variables then, this equation has the two results that are:
a = -1 or a = 2
Now I am going in deep with the topic so the next topic is, How to evaluate an algebraic expression? To evaluate or to explore an algebraic expression refers to substitute or place specific values (desired values) for its variables. Let’s take a simple example of an algebraic expression to understand it better:
algebraic = 2a + 1
Now we are going to evaluate the given algebraic expressions. Now what we need to do is to try various values of variable which satisfy the given algebraic expression. Lets take a = 3, which provides us: algebraic = 2 x 3 + 1 = 7. we can say that the value of algebraic at a = 3 is 7. let’s take an example which is little tougher or a bit complex then the above one
Now if we use above algebraic and put value of a = 2b + 1, where b is another variable which gives
Algebraic = 2 x (2b + 1) + 1 = 4b + 2 + 1 = 4b + 3
Now we can further solve this by substituting different values for variable b.
Before proceeding further towards inequalities let’s talk about equivalent expressions: Two expressions are equivalent if their values are equal for all possible evaluations of the two expressions. In other words presenting them with an equality sign between the expressions gives an identity.
Now I am going to talk about eighth grade linear equations topic:
In simple mathematical manner we can say that any equation that when graphed produces a straight line, then the equation is called as Linear Equation or we can say that any equation is a linear if it can be written in the linear form : ax = b. Here x is the variable, a and b are the constants. The common form of a linear equation in the two variables like x and y is
y = mx + b
where x and y are two variables and m and b are two constants.
The constant m determines the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b is also known as Y-Intercept.
There are three possible solutions for the linear equations:
Unique Solutions: if a not equal to b then only possible solution: x = b/a
No Solution: If a = 0 and b is not equal to 0 then it has no solution.
Infinite Solutions or many solutions: if a = 0 and b = 0. In this solution 0x = 0 and there are infinite solutions for all the values of x.
Let’s take an example : 2a + 3 = a + 5
For solving this we need to subtract 3 from both the sides 2a = a + 2 that will result in a = 2.
The most important thing to understand is that, for solving a linear equation we need to recognize that the equation is linear or non linear and if linear then convert it to the simplest form. Lets take an example to elaborate it well:
(a – 2)(a – 3) = (a + 1)(a + 2)
what we need to do is to apply distributive property to both the sides of the equation
axa – ax3 – ax2 + 2x3 = axa + 2xa + 1xa + 1x2
on further simplifying this we get
a2 – 5a + 6 = a2 + 3a + 2
This is what we get, we can further simplify it by subtracting a2 from both the sides.
Now after telling the basic concept behind linear equations, I am going to discuss about inequalities for class eighth. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. The most important thing to understand is the use of inequality symbols. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≤ less than or equal to etc.
In eighth standard we learn about linear inequality. So what linear inequality means is the first query comes in our mind. A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything on One side of a line on a graph. In mathematics a linear inequality is an inequality which involves a linear function.
The most important thing is to understand, how the inequality sign reverse when negative value comes. So to understand linear inequality, take an example: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12
-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the two.
We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.
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