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Algebraic Expressions and Algebra basics
Post On. Algebra and algebraic expressions can become a nightmare for those who fear! But it can be as magical as ‘abracadabra’ and you can have fun with these equations and expressions altogether. Let’s see what this algebraic expressions is and how it works.
1. What is algebra Expressions?
2. What is algebra Equation?
3. How to solve an algebraic equation?
4. Why do we need algebraic equations?
5. But Algebra is not as simple as this!
6. Linear Algebra: Linear equations and Inequality
7. What is System of Equations?
8. How to solve system of equations?
9. Advanced Algebra: Quadratics and Polynomials
10. Important questions asked by students related to algebraic expressions
Expression meaning:- Consider any expression, say, 2+4; we need to check if it fits the criteria of the definition of the algebraic expressions. An algebraic expressions should have a constant value, here, it is 2. It should have a function, in this case, it is ‘+’ or expression is a combination of variable and Constant Now, an algebraic expressions should also have a variable. In this case what we can do is re-write the expression as 2+4x, with the value of x being 1. So, a real algebraic expressions can be written as, 2+4x and it will be called an algebraic expressions.
When the sign of = comes in the picture, an expression becomes an equation. So, the expression 2+4x=6 becomes an algebraic equation.
The basic step to solving any algebraic equation is to get the like terms together first and then proceed with the functions. In our equation, 2+4x=6, we can see that there are two constants 2 and 6. Let
us try solving this simple equation.
2+4x=6
4x=6-2
4x=4
x=1
See, this is what we considered as the value of ‘x’ in the first place.
Well, algebra comes from the Arabic word ‘al jabr’ and you know what it means, to bring the broken pieces together’. So, you can very well guess that algebra holds great importance in life as well because
we may not use terms like ‘variables’, ‘constants’, or ‘x’ and ‘y’, but we do need to bring many broken pieces together to make something meaningful every now and then.
The basic use of algebra is
finding logic. When you try to predict any possibility or calculate any future event, you are doing algebra. Something as simple as catching the ball is actually solving an algebraic equation. So even
a toddler who is learning to catch things is unknowingly solving an algebraic equation.
While making the monthly budget of your groceries, you are using algebraic equations. For example, your
budget is 500, you need to buy 3 items. How will you predict the expenditure? You will do it with an algebraic equation.
500 = x+y+z where x, y, and z are the individual cost of three items. Now,
according to the quantity you need to buy for each item, you will divide the amount you wish to spend on each item according to your priority.
And when it comes to professional use, algebra is the
basis of any programming language, business analysis, and profit and loss calculations, the motion of planets, devising trajectories for satellites, and much more. So, you need to learn your algebra
lessons very thoroughly if you wish to have a career as a computer programmer or a business analyst, or a scientist or astronomer, how cool is that!
Yes, of course it is. Solving algebraic equations can be aided with linear equations, graphs and illustrations that makes it easy to understand.
This topic is extremely important as it deals with the geometry of both basic and advanced levels. You need to be efficient in linear algebra to model anything, be it the structure of a house, a model of
a machine, computing trajectories, etc.
Linear algebra helps you solve equations with the help of two or three variables, 2-dimensional and 3-dimensional respectively
Let’s take the above example
for understanding. 2+4x=6, This solution is just for one value of x, which is x=1. If we need to find solutions for different values of ‘x’, the solution won’t be 6 anymore. So, the equation will change
to 2+4x=2y. Here, ‘y’ is another variable whose value depends upon the value of ‘x’. This forms a linear equation.
We need to reconstruct the equation in order to find the value of ‘y’ in terms
of ‘x’. So, let us reconstruct it.
2+4x= 2y
y= (2+4x)/2
y= 1+2x
This equation forms the base of graphing lines and slopes. With the help of slopes, you can solve problems related to positions of objects in motions,
time and distance, heights and depths. These come in handy in astronomy, geography, archeology, architecture, defense, military, and many other fields.
Now, when there are more than one variables in a problem and you need to find out the value of all the variables, it won’t be possible with the help of just one equation, you will need a set of equations
to satisfy values of each variable. It is called the system of equations. Let us again consider the first example. 1+2x=y is what we got. Now, if we have to find the value of both ‘x’ and ‘y’, we need
to have another equation with the same variables. So,
The first equation is: y=1+2x
The second equation is: y=1+2x
To solve this system of equations, again we go with the basic approach of putting similar things together. So, eliminate or substitute one variable from an equation. Let us replace ‘y’ in this case.
y=1+2x and y=3+6x also. So, let us replace the value of ‘y’ from the second equation into the first one. The first equation will become,
y = 1+2x
3+6x =1+2x
Now, this equation becomes easy to solve, right?
6x-2x = 1-3
4x = (-2)
x= (-1/2)
Wait, this doesn’t end here. Now, you have to replace the value of ‘x’ in any of the two equations and find the value of ‘y’, pretty easy, right?
Put the value of ‘x’ in first equation,
y=1+2x
y=1+2*(-1/2)>
y=0
I am absolutely sure that by now, you have become comfortable with the basics of algebra you wish to delve deeper into the subject. What we discussed was just about linear equations and there is much more to it. The equations may change from linear to non-linear and there can arise terms like quadratic equations where you have to deal with squares, cubes and higher degree powers of variables to find the solutions. Interesting, isn’t it? Find out yourself and see if you like your algebra! Happy mathematics!
Question.1. Most used mathematics formulas and simplification formulas in algebraic expressions?
Answer:-
1. (a + b)2 = a2 + 2ab + b2
2. (a – b)2 = a2 – 2ab + b2
3. a2 – b2 = (a + b)(a – b)
4. (x + a)(x + b) = x2 + (a + b) x + ab
5. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
6. (a + b)3 = a3 + b3 + 3ab (a + b)
7. (a – b)3 = a3 – b3 – 3ab (a – b)
8. a3 + b3 + c3– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
Question:-2.What is algebraic expressions example?
Answer:-
Consider any expression, say, 2+4; we need to check if it fits the criteria of the definition of the algebraic expressions. An algebraic expressions should have a constant value, here, it is 2. It should
have a function, in this case it is ‘+’.
Now, an algebraic expressions should also have a variable. In this case what we can do is re-write the expression as 2+4x, with the value of x being 1. So,
a real algebraic expressions can be written as, 2+4x and it will be called an algebraic expressions.
Question:-3.How to solve algebraic questions?
Answer:-
To solve this system of equations, again we go with the basic approach of putting similar things together. So, eliminate or substitute one variable from an equation. Now see simplification question
Let
us replace ‘y’ in this case.
y=1+2x and y=3+6x also. So, let us replace the value of ‘y’ from the second equation into the first one. The first equation will become,
y=1+2x
3+6x=1+2x
Now, this equation becomes easy to solve, right?
6x-2x = 1-3
4x = (-2)
x= (-1/2)
Wait, this doesn’t end here. Now, you have to replace the value of ‘x’ in any of the two equations
and find the value of ‘y’, pretty easy, right?
Put the value of ‘x’ in first equation,
y=1+2x
y=1+2*(-1/2)
y=0
Question:-4. How do you identify algebraic expressions and identities?
Answer:-
When the sign of = comes in picture, an expression becomes an equation. So the expression 2+4x=6 becomes an algebraic equation.
What is algebra Expressions?
Expression meaning:- Consider any expression, say, 2+4; we need to check if it fits the criteria of the definition of the algebraic expressions. An algebraic expressions should have a constant value, here, it is 2. It should have a function, in this case, it is ‘+’ or expression is a combination of variable and Constant Now, an algebraic expressions should also have a variable. In this case what we can do is re-write the expression as 2+4x, with the value of x being 1. So, a real algebraic expressions can be written as, 2+4x and it will be called an algebraic expressions.
What is algebra Equation?
When the sign of = comes in the picture, an expression becomes an equation. So, the expression 2+4x=6 becomes an algebraic equation.
How to solve an algebraic equation?
The basic step to solving any algebraic equation is to get the like terms together first and then proceed with the functions. In our equation, 2+4x=6, we can see that there are two constants 2 and 6. Let us try solving this
simple equation.
2+4x=6
4x=6-2
4x=4
x=1
See, this is what we considered as the value of ‘x’ in the first place.
Why do we need algebraic equations?
Well, algebra comes from the Arabic word ‘al jabr’ and you know what it means, to bring the broken pieces together’. So, you can very well guess that algebra holds great importance in life as well because we may not use terms
like ‘variables’, ‘constants’, or ‘x’ and ‘y’, but we do need to bring many broken pieces together to make something meaningful every now and then.
The basic use of algebra is finding logic. When you try to predict
any possibility or calculate any future event, you are doing algebra. Something as simple as catching the ball is actually solving an algebraic equation. So even a toddler who is learning to catch things is unknowingly
solving an algebraic equation.
While making the monthly budget of your groceries, you are using algebraic equations. For example, your budget is 500, you need to buy 3 items. How will you predict the expenditure?
You will do it with an algebraic equation.
500 = x+y+z where x, y, and z are the individual cost of three items. Now, according to the quantity you need to buy for each item, you will divide the amount you wish to spend
on each item according to your priority.
And when it comes to professional use, algebra is the basis of any programming language, business analysis, and profit and loss calculations, the motion of planets, devising
trajectories for satellites, and much more. So, you need to learn your algebra lessons very thoroughly if you wish to have a career as a computer programmer or a business analyst, or a scientist or astronomer, how cool
is that!
But Algebra is not as simple as this!
Yes, of course, it is. Solving algebraic equations can be aided with linear equations, graphs, and illustrations that make it easy to understand.
Linear Algebra: Linear equations and Inequality
This topic is extremely important as it deals with the geometry of both basic and advanced levels. You need to be efficient in linear algebra to model anything, be it the structure of a house, a model of a machine, computing
trajectories, etc.
Linear algebra helps you solve equations with the help of two or three variables, 2-dimensional and 3-dimensional respectively
Let’s take the above example for understanding. 2+4x=6, This solution
is just for one value of x, which is x=1. If we need to find solutions for different values of ‘x’, the solution won’t be 6 anymore. So, the equation will change to 2+4x=2y. Here, ‘y’ is another variable whose value depends
upon the value of ‘x’. This forms a linear equation.
We need to reconstruct the equation in order to find the value of ‘y’ in terms of ‘x’. So, let us reconstruct it.
2+4x= 2y
y= (2+4x)/2
y= 1+2x
This equation forms the base of graphing lines and slopes. With the help of slopes, you can solve problems related to positions of objects in motions, time and distance,
heights and depths. These come in handy in astronomy, geography, archeology, architecture, defense, military, and many other fields.
What is System of Equations?
Now, when there are more than one variables in a problem and you need to find out the value of all the variables, it won’t be possible with the help of just one equation, you will need a set of equations to satisfy values
of each variable. It is called the system of equations. Let us again consider the first example. 1+2x=y is what we got. Now, if we have to find the value of both ‘x’ and ‘y’, we need to have another equation with the
same variables. So,
The first equation is: y=1+2x
The second equation is: y=1+2x
How to solve system of equations?
To solve this system of equations, again we go with the basic approach of putting similar things together. So, eliminate or substitute one variable from an equation. Let us replace ‘y’ in this case.
y=1+2x and y=3+6x
also. So, let us replace the value of ‘y’ from the second equation into the first one. The first equation will become,
y = 1+2x
3+6x =1+2x
Now, this equation becomes easy to solve, right?
6x-2x = 1-3
4x = (-2)
x= (-1/2)
Wait, this doesn’t end here. Now, you have to replace the value of ‘x’ in any of the two equations and find the value of ‘y’, pretty easy, right?
Put the value of ‘x’ in first equation,
y=1+2x
y=1+2*(-1/2)>
y=0
Advanced Algebra: Quadratics and Polynomials
I am absolutely sure that by now, you have become comfortable with the basics of algebra you wish to delve deeper into the subject. What we discussed was just about linear equations and there is much more to it. The equations may change from linear to non-linear and there can arise terms like quadratic equations where you have to deal with squares, cubes and higher degree powers of variables to find the solutions. Interesting, isn’t it? Find out yourself and see if you like your algebra! Happy mathematics!
Answer:-
1. (a + b)2 = a2 + 2ab + b2
2. (a – b)2 = a2 – 2ab + b2
3. a2 – b2 = (a + b)(a – b)
4. (x
+ a)(x + b) = x2 + (a + b) x + ab
5. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
6. (a + b)3 = a3 + b3 + 3ab
(a + b)
7. (a – b)3 = a3 – b3 – 3ab (a – b)
8. a3 + b3 + c3– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
Answer:-
Consider any expression, say, 2+4; we need to check if it fits the criteria of the definition of the algebraic expressions. An algebraic expressions should have a constant value, here, it is 2. It should have a
function, in this case it is ‘+’.
Now, an algebraic expressions should also have a variable. In this case what we can do is re-write the expression as 2+4x, with the value of x being 1. So, a real algebraic
expressions can be written as, 2+4x and it will be called an algebraic expressions.
Answer:-
To solve this system of equations, again we go with the basic approach of putting similar things together. So, eliminate or substitute one variable from an equation. Now see simplification question
Let us replace
‘y’ in this case.
y=1+2x and y=3+6x also. So, let us replace the value of ‘y’ from the second equation into the first one. The first equation will become,
y=1+2x
3+6x=1+2x
Now, this equation becomes easy to solve, right?
6x-2x = 1-3
4x = (-2)
x= (-1/2)
Wait, this doesn’t end here. Now, you have to replace the value of ‘x’ in any of the two equations
and find the value of ‘y’, pretty easy, right?
Put the value of ‘x’ in first equation,
y=1+2x
y=1+2*(-1/2)
y=0
When the sign of = comes in picture, an expression becomes an equation. So the expression 2+4x=6 becomes an algebraic equation.
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