# Algebraic Expressions and Algebra basics

*Post On.*04-Dec-2019

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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.

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.

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!

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.

The first equation is: y=1+2x

The second equation is: y=1+2x

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.1. Most used mathematics formulas and simplification formulas in algebraic expressions?**

**Answer:-**

1. (a + b)^{2} = a^{2} + 2ab + b^{2}

2. (a – b)^{2} = a^{2} – 2ab + b^{2}

3. a^{2} – b^{2} = (a + b)(a – b)

4. (x + a)(x + b) = x^{2} + (a + b) x + ab

5. (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

6. (a + b)^{3} = a^{3} + b^{3} + 3ab (a + b)

7. (a – b)^{3} = a^{3} – b^{3} – 3ab (a – b)

8. a^{3} + b^{3} + c^{3}– 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.

**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.

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.

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 make it easy to understand.

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.

The first equation is: y=1+2x

The second equation is: y=1+2x

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

**Answer:-**

1. (a + b)^{2} = a^{2} + 2ab + b^{2}

2. (a – b)^{2} = a^{2} – 2ab + b^{2}

3. a^{2} – b^{2} = (a + b)(a – b)

4. (x
+ a)(x + b) = x^{2} + (a + b) x + ab

5. (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

6. (a + b)^{3} = a^{3} + b^{3} + 3ab
(a + b)

7. (a – b)^{3} = a^{3} – b^{3} – 3ab (a – b)

8. a^{3} + b^{3} + c^{3}– 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

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