 # Algebraic Equations

Let us begin with having an insight into Algebraic equation- An algebraic equation is basically defined as a mathematical statement in which two expressions are set equal to each other. In other words, equations mean equality i.e. the equal sign. This is the basis for what algebra equations are all about- “equating one quantity with another”. Equations are very much similar to a balance scale. A balance scale would indicate that an equal amount of weight needs to be placed on either side in case of any mismatch causing the imbalance. Imagine if we add some weight to just one side. The scale will tilt on one side. Algebra equations follow the same logic. The final value has to be the same on both sides else becomes an inequality.

### Algebraic Equations Examples

An equation says that two things are equal. This means that the equation will always have an equal sign ‘=’ like this:

x+5=10

The above equation says that both sides are balanced: what is on the left (x+5) is equal to what is on the right (10). The equation is balanced as both sides have the same value. If you solve the equation, the value of x will be 5, the equation will be mathematically balanced. To avoid committing an error that tips the equation out of balance, ensure that any change on either side of the equation is reciprocated on the opposite side. For example, if you want to add a number 2 to the left side of the equation you will have to add the same 2 to the other side of the equation i.e.

x+5=10

x+5+2=10+2

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

### Types of Algebraic Equations

Algebraic equations questions are solved based on their position of variables, the functions and the types of operators used. A few of the equations in algebra are discussed below.

Polynomial Equations: All polynomial equations are different algebraic equations, but all algebraic equations are not polynomial equations. To recall the basics, a polynomial equation is an equation that consists of variables, exponents and coefficients.

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

Quadratic Equations: A quadratic equation is a polynomial equation of degree 2 in one variable of type.

The quadratic equations are polynomials with degree 2 in one variable of type  f(x) = ax² + bx + c.

Quadratic Equations: ax²+bx+c=0 (a not equal to 0)

Cubic Equations: A cubic equation is a polynomial equation of degree 2. All the cubic polynomials are also algebraic equations.

Cubic Polynomials: ax³+bx²+cx+d=0

Rational Polynomial Equations

P(x)/Q(x)=0

Trigonometric Equations: All the trigonometric equations are considered as algebraic functions. For a trigonometry equation, the expression includes the trigonometric functions of a variable.

Trigonometric Equations: cos2x = 1+4sinx

### How to Solve Algebraic Equations?

Consider the following situation. I am going to the market. In one carry bag, I carry some potatoes, onions and papayas. The bag can carry a total of 8 items in it. So I purchased 4 potatoes and 2 onions. How many papayas can I now carry?

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

4 potatoes + 2 onions + ‘x’ papaya = 8 items

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

Now, let’s solve this equation:

4+2+x=8

6+x=8

6+x-6=8−6

x=2

I can carry 2 papayas for my trip.

In the same way, what would certainly represent an inequality? Definitely, when the left-hand side is not equal to the right-hand side. How would this happen?

Taking the same example, 6 + x = 8, and change that equal to a greater than or a lesser than sign. They aren't equations anymore. Look for some more examples to clarify this concept.

x + 5 = 10 is an equation

xy + 3 = z is an equation

But, 6p < 77 is not.

Let’s see some of the algebraic equations examples with answers

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

Solution

Given equation: 2(p+4)+3(p−5)–2q=0

2p+2×4+3p–3×5–2q=0 (Using Distributive property to get rid of parenthesis)

2p+8+3p–15–2q=0 (Simplifying).

5p–2q–7=0 (on further simplifying terms).

1. What is the difference between expression and equations?

When numbers, variables and operators are clubbed together to show the value of something is called an expression, whereas an equation is described as a mathematical statement consisting of two expressions equal to one another.

There is no relation symbol in an expression. On the other hand, an equation is indicated by an equal sign (=). An expression is always one-sided and it has no left or right side. Unlike, an equation is two-sided, there is an equal sign that separates the left and right sides.

The answer to an equation is only true or false. As opposed to the expression, which could be either an expression or a numerical value.

2. What are the different components of the Algebra formulas and expressions?

The Algebra equations are represented as  formulas and expressions, which in turn are combinations of numbers, variables and at least one arithmetic operation

For example, 5(x-2)+9y-4

Factors: A factor can be a variable, number, term, or even a long tail expression. In the above equation, 5 is a factor of (x-2) and vice versa

Coefficient: In the example given 9 is the coefficient of y. Hence, the coefficient is always a numerical factor of an expression that contains a variable.

Variable: From the word itself, you can understand a variable does not have a fixed value but varies depending upon the change in expression. For example, y is the variable of 9 in the given example.