Friends today we all are going to start our eighth grade mathematics and I am going to discuss about Algebra section. Before proceeding further let's talk about the basic concepts of Algebra. Algebra is a very vast area of study of mathematics which almost covers 90 percent of mathematics. In simple mathematical manner we can say that it is a branch of mathematics which deals with the study of the rules of operations and relations. Equation term can be explained as a mathematical expressions which shows the equality of two expressions. For example b + z = 6. In most general manner we can say that any combination of literal symbols and numbers that results from algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting a root) are called as Algebraic equations. The two most important types of such Algebraic equations are linear equations which is written in the form y = mx + b, and quadratic equations that can be represented in the form y = ax2 + bx + c. In general Algebraic equations are useful for modeling real life phenomena.
Following steps are used to simplify an algebraic equation. Firstly remove all the fractions in the equation Then remove the parentheses . Combine all the like terms so that we get all the variables and terms together. Move all the variable terms by adding or subtracting on both sides of the equal sign so the variable terms are all on one side of the equal sign. And finally if there is any multiplication sign then remove it by dividing.
Let's take an example to understand it better. The problem is to find the product of the following algebraic equations:
(b + 5)(b – 3) here multiply each multinomial term to the another multinomial term like
b x b – 3 x b + 5 x b – 3 x 5 = b2 + 2b – 15
Now let's take the first sub topic of eighth grade Algebra problem that is Proportional and non proportional linear relationships. First question comes in our mind is what is a proportion ? In simplest of manner we can say that forming a relationship with other parts or quantities is called as proportion. Relationship between the variables is basically a way in which the variables change. The relationship can be linear, directly proportional and non linear.
Before proceeding further let's talk about Rational expression. Rational expression is an expression which can be written as a fraction a/b. Here a is the numerator and b is a denominator. The most important thing to understand is that denominator can never be zero.
Let's take an example:
The numbers 5/3 and -6/11 are rational numbers.
The number 5 is also an expression : 5/1 = 5 and any number in decimal form also an expression for example: 3.33 is a rational number : 3.33 = 333/100.
Some theorems which tell about Rational Expressions:
First one is that any integer is a rational. For example a number n = n/1.
Second: the representation of rational number as a fraction is not unique. Like 3/4 = 6/8 = -9/-12.
Third : Every nonzero rational number or a rational that do not contain a 0 has a representation in lowest term.
Closure Property of Rational Number shown as:
Adding and Subtracting of a fraction is done by using this property: x/y + a/b = xb + ay / yb.
For Multiplying a fraction this property comes in an account: a/b x c/d = ac/bd
and for dividing a fraction we use this property: a/b divide c/d = ad/bc.
For simplifying rational expression, we must need to have good factoring skills. It requires two steps in solving a rational expressions.
factor the numerator and denominator is the first step and the second step is divide all common factors that the numerator and denominator have.
Dividing a rational number is the most difficult part as it requires key skills. Now we are going to learn how can we divide a rational expression:
12/5 divide by9/5 then we need to take a reciprocal of 9/5 . The reciprocal of 9/5 is 5/9.
Multiply 12/5 with the reciprocal. 12/5 x 5/9 = 12/9 = 4/3.
Linear relationship can be stated that the two variables always form a straight line when graphed. The most common example to understand this is the distance time graphs of the stationary object and any of the object which is moving with the constant velocity. Directly proportional can be defined as the rate of increase in one variable is similar to the rate of increase in the other variable. The example to understand: force is directly proportional to its mass. Non Linear relationship means that the rate of increase in one variable is distinct from the rate of increase in the other variable.
Let's discuss about Non Proportional linear relationship. This can be explained in the general form of linear expression that is y = mx + b, Here b is not equal to zero, m is the slope of the line or we can say that it represents the constant rate of change, b = Y intercept form. The graph of a non proportional linear relationship is a straight line which can never pass through the origin.
Let's take an example to understand the Non proportional linear relationships (y = mx + b, b not equal to 0).
The taxi provider company charges a flat fee of Rs. 50 plus Rs. 30 per mile to ride in a taxi.
Assumed that the flat fee is acquired as soon as the person enters the taxi.
What we need to do is to find out the cost of the taxi ride, multiply the total number of miles traveled by Rs. 30 and then add it with Rs. 50 (Rs. 50 is the flat fee) to the product.
If we are going to simplify the above equation then we need to follow this : If y shows the total cost of a taxi ride of x miles, then the relationship can be represented as an equation in the form of y = mx + b, here m represents the cost per mile (Rs. 30/mile) and b represents the flat fee (Rs. 50).
Total Cost = Cost per mile * Number of Miles + Flat Fee
Y = 30 * X + 50, or we can also make it y = 30x + 50
Another example to show Non-proportional Linear relationship with negative slope. The problem is that a ten inch candle burns at a constant rate of one inch per hour.
What we need to do: To find out the exact height of the candle you need to multiply the number of hours that the candle burns by 1 inch per hour, and than subtracting the product from the candle's initial height that is ten inches.
Solution : Similarly if y shows the height of the candle after x hours of burning, then the relationship can be explained in the form of an equation that is Y = mx + b, here m represents the rate at which the candle burns which is I inch per hour and b shows the initial height of the candle which is 10 inches.
Height of Candle = Rate at which it Burns * Number of Hours Burned + Initial Height
y = -1 * x + 10, or it can make it y = 10 -1x
We can explain proportional relationship by the following methodology. Let's take a general method to understand it. In any of the relationship let's take between y and z is proportional, it means that as y changes , x also changes by the same percentage. It shows that if y grows by 20 percent of y, z also grows by 20 percent of z. In an algebraic form we have already discussed it that is y = mx, where m is a constant.
Directly proportional means : A very common delusion is that two variables are directly proportional if one increases as the other increases or vice versa. Another thing is that two variables are explained to be directly proportional if and only if their ratio is a constant for all the values of each variable. Thus the most important outcome is that one variable is divided by the other , the answer is always a constant.
Now the next topic we all are going to understand in next section is Sequences: A sequence is basically a systematic listing of objects. In mathematical world the list formed from sequence can be numbers, algebraic expressions etc. If the objects in the list are numbers then it is a numeric sequence. There are many types of sequences in mathematics but the most common types of sequences are:
Arithmetic Sequence
Geometric Sequence
Harmonic Sequence
In next class we will discuss sequences in detail and also going to learn about the types of sequences in mathematics.
Following steps are used to simplify an algebraic equation. Firstly remove all the fractions in the equation Then remove the parentheses . Combine all the like terms so that we get all the variables and terms together. Move all the variable terms by adding or subtracting on both sides of the equal sign so the variable terms are all on one side of the equal sign. And finally if there is any multiplication sign then remove it by dividing.
Let's take an example to understand it better. The problem is to find the product of the following algebraic equations:
(b + 5)(b – 3) here multiply each multinomial term to the another multinomial term like
b x b – 3 x b + 5 x b – 3 x 5 = b2 + 2b – 15
Now let's take the first sub topic of eighth grade Algebra problem that is Proportional and non proportional linear relationships. First question comes in our mind is what is a proportion ? In simplest of manner we can say that forming a relationship with other parts or quantities is called as proportion. Relationship between the variables is basically a way in which the variables change. The relationship can be linear, directly proportional and non linear.
Before proceeding further let's talk about Rational expression. Rational expression is an expression which can be written as a fraction a/b. Here a is the numerator and b is a denominator. The most important thing to understand is that denominator can never be zero.
Let's take an example:
The numbers 5/3 and -6/11 are rational numbers.
The number 5 is also an expression : 5/1 = 5 and any number in decimal form also an expression for example: 3.33 is a rational number : 3.33 = 333/100.
Some theorems which tell about Rational Expressions:
First one is that any integer is a rational. For example a number n = n/1.
Second: the representation of rational number as a fraction is not unique. Like 3/4 = 6/8 = -9/-12.
Third : Every nonzero rational number or a rational that do not contain a 0 has a representation in lowest term.
Closure Property of Rational Number shown as:
Adding and Subtracting of a fraction is done by using this property: x/y + a/b = xb + ay / yb.
For Multiplying a fraction this property comes in an account: a/b x c/d = ac/bd
and for dividing a fraction we use this property: a/b divide c/d = ad/bc.
For simplifying rational expression, we must need to have good factoring skills. It requires two steps in solving a rational expressions.
factor the numerator and denominator is the first step and the second step is divide all common factors that the numerator and denominator have.
Dividing a rational number is the most difficult part as it requires key skills. Now we are going to learn how can we divide a rational expression:
12/5 divide by9/5 then we need to take a reciprocal of 9/5 . The reciprocal of 9/5 is 5/9.
Multiply 12/5 with the reciprocal. 12/5 x 5/9 = 12/9 = 4/3.
Linear relationship can be stated that the two variables always form a straight line when graphed. The most common example to understand this is the distance time graphs of the stationary object and any of the object which is moving with the constant velocity. Directly proportional can be defined as the rate of increase in one variable is similar to the rate of increase in the other variable. The example to understand: force is directly proportional to its mass. Non Linear relationship means that the rate of increase in one variable is distinct from the rate of increase in the other variable.
Let's discuss about Non Proportional linear relationship. This can be explained in the general form of linear expression that is y = mx + b, Here b is not equal to zero, m is the slope of the line or we can say that it represents the constant rate of change, b = Y intercept form. The graph of a non proportional linear relationship is a straight line which can never pass through the origin.
Let's take an example to understand the Non proportional linear relationships (y = mx + b, b not equal to 0).
The taxi provider company charges a flat fee of Rs. 50 plus Rs. 30 per mile to ride in a taxi.
Assumed that the flat fee is acquired as soon as the person enters the taxi.
What we need to do is to find out the cost of the taxi ride, multiply the total number of miles traveled by Rs. 30 and then add it with Rs. 50 (Rs. 50 is the flat fee) to the product.
If we are going to simplify the above equation then we need to follow this : If y shows the total cost of a taxi ride of x miles, then the relationship can be represented as an equation in the form of y = mx + b, here m represents the cost per mile (Rs. 30/mile) and b represents the flat fee (Rs. 50).
Total Cost = Cost per mile * Number of Miles + Flat Fee
Y = 30 * X + 50, or we can also make it y = 30x + 50
Another example to show Non-proportional Linear relationship with negative slope. The problem is that a ten inch candle burns at a constant rate of one inch per hour.
What we need to do: To find out the exact height of the candle you need to multiply the number of hours that the candle burns by 1 inch per hour, and than subtracting the product from the candle's initial height that is ten inches.
Solution : Similarly if y shows the height of the candle after x hours of burning, then the relationship can be explained in the form of an equation that is Y = mx + b, here m represents the rate at which the candle burns which is I inch per hour and b shows the initial height of the candle which is 10 inches.
Height of Candle = Rate at which it Burns * Number of Hours Burned + Initial Height
y = -1 * x + 10, or it can make it y = 10 -1x
We can explain proportional relationship by the following methodology. Let's take a general method to understand it. In any of the relationship let's take between y and z is proportional, it means that as y changes , x also changes by the same percentage. It shows that if y grows by 20 percent of y, z also grows by 20 percent of z. In an algebraic form we have already discussed it that is y = mx, where m is a constant.
Directly proportional means : A very common delusion is that two variables are directly proportional if one increases as the other increases or vice versa. Another thing is that two variables are explained to be directly proportional if and only if their ratio is a constant for all the values of each variable. Thus the most important outcome is that one variable is divided by the other , the answer is always a constant.
Now the next topic we all are going to understand in next section is Sequences: A sequence is basically a systematic listing of objects. In mathematical world the list formed from sequence can be numbers, algebraic expressions etc. If the objects in the list are numbers then it is a numeric sequence. There are many types of sequences in mathematics but the most common types of sequences are:
Arithmetic Sequence
Geometric Sequence
Harmonic Sequence
In next class we will discuss sequences in detail and also going to learn about the types of sequences in mathematics.
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