Algebraic Equations

What comes to mind when you hear the term “equation”? Equations… equality… equal sign? That’s what equations are all about. Equating one quantity with another.

Equations In Algebra

Equations are like a balance scale. If you’ve seen a balance scale, you would know that an equal amount of weight has to be placed on either side for the scale to be considered “balanced”. If we add some weight to just one side, the scale will tip on one side and the two sides are no longer in balance. Equations follow the same logic. Whatever is on one side of the equal sign must have exactly the same value on the other side else it becomes an inequality.

Equations

For example, in an equation 1+1 = 2, it is balanced as both sides have the same value. To avoid committing an error that tips the equation out of balance, make sure that any change on one side of the equation is reciprocated on the other side. For example, if you want to add a number 5 to one side of the equation you will have to add the same 5 to the other side of the equation.

1 + 1 = 2

1 + 1 + 5 = 2 + 5

The same goes for subtraction, multiplication, and division. As long as you do the same thing to both sides of the equation it will remain balanced.

Consider the following situation. I am going for a trip. In one bag I carry some t-shirts, shorts, and towels. A total of 8 items can fit in the bag. So I pack 4 shirts and 2 shorts. How many towels can I now carry?

Consider the number of towels to be ‘x’. Let’s form the equation now.

4 shirts + 2 shorts + ‘x’ towels = 8 clothes

The left-hand side (LHS) of our equation is being compared to the right-hand side (RHS) of the equation.

Many a times students are confused between expressions and equation. Here is the difference between them

Algebr - Expressions And Equations

Solved Examples

Example– Let’s solve this equation.

4+2+x=8

6+x=8

6+x-6=8-6

x=2

I can carry 2 towels for my trip.

In the same way, what would depict an inequality? Obviously,when the left hand side is not equal to the right hand side. How would this happen?

Let’s take the same 6 + x = 8 and change that equal to into a greater than or a lesser than sign. These aren’t equations! Consider some examples to clarify this concept.

x + 2 = 21, xy + 9 = z are equations but 6p > 77 is not.

Algebraic equations are of various types, some of them being:

Polynomial equations-
Linear equations: ax+b=c (a not equal to 0)

Quadratic equations

\(ax^{2} + bx + c = 0\) (a not equal to 0)

Cubic equations-

\(ax^{3} + bx^{2} + cx + d = 0\)

Rational polynomial equations-

P(x)/Q(x)=0

Trigonometric equations-
cos2x = 1+4sinx

Solving algebraic equations:

Example: Simplify the given equation : \(2 (x+4) +3 (x – 5) – 2y =0\)

Solution: Given equation: \(2 (x+4) +3 (x – 5) – 2y = 0\)

\(2x + 2 \times 4 + 3x – 3 \times 5 – 2y = 0\)   (Using Distributive property to get rid of parenthesis)

\(2x + 8 + 3x – 15 – 2y = 0 \)    (Simplifying)

\(5x – 2y – 7 = 0\)     (on further simplifying terms)

Algebraic Expressions

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