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How to Solve an Equation?

equations

In mathematics, you must have come across the word ‘equations’. You must have read questions like, ‘Solve the equation 2x + 12 = 33 to find the value of x’. But, what is the meaning of the term equation? Equations, in math, are statements. These statements often have two linked expressions with an ‘equal to sign. Equations can even have more than two expressions. Mathematicians have solved some equations having more than two expressions up to a certain level. Beyond that, even they are rendered helpless. But, we don’t have to worry about those difficult equations. In this article, we will learn the fundamental concepts of every equation known to humanity. We will study what an equation is and how many types of equations are there in this world.  

Equations Definition

Equations are very easy to identify. Wherever you see two algebraic expressions on the left and right-hand sides of the equal ‘=’ sign, you have found an equation. So we can say that statements with the ‘equal to’ sign in between them are known as equations. If the equal to symbol is not present between the expressions, they are not considered an equation. In the next section of this article, we will see the difference between equations and expressions. For now, look at the examples given below:

Example 1: Is x + y = 13 an equation?

Solution: Yes, x + y = 13 is an equation because it has an ‘equal to’ sign between x + y and 13.

Example 2: Is a – 24 + 13y an equation?

Solution: No! a – 24 + 13y is not an equation because it doesn’t have an equal sign.

Example 3: Is 66 – 12 = 34 – 3 an equation?

Solution: Indeed, this is an equation because 66 – 12 and 34 – 3 have the ‘equal to’ sign between them.

Equations are used in mathematics to solve the values of unknown quantities. They are used in solving word problems and many daily life problems involving time and distance, work, profit and loss, and many more. Let us now learn the basic difference between an equation and an expression.

Equation Vs. Expression

Equations and expressions are two different things. Students often confuse the two and make a lot of mistakes. See the table mentioned below to clarify the difference between an equation and an expression.

EquationExpression
In math, an equation is formed when two expressions have the same value and are put together with an ‘equal to’ symbol in the middle.It’s a mathematical statement with at least one word and several terms linked by operators.
It has a “=” symbol next to it.An equal to “=” symbol does not appear in an expression.
It can be solved to determine the unknown quantity’s value.Maximum it can be reduced to its most basic form.
For example: 7x – 2a = 34, 4a + 22 = 3c + b, etc.For example: 34x – 3z + 2y, a + 3k, etc.

Parts of an Equation

In every equation, the left-hand side should be equal to the right-hand side. Both sides must be equal. Coefficients, variables, operators, constants, terms, expressions, and an ‘=’ sign are all essential components of an equation. An equation may have any one or all of these terms revolving around the ‘=’ sign. Let us learn these terms individually:

  • Term: Any numeric or algebraic entity present around the operators is known as a term. 
  • Variables: All the alphabetic terms present in an equation whose value is unknown are known as variables.
  • Constants: All the numeric terms of an equation are known as constants.
  • Coefficients: The constant terms present just before the variables are coefficients. 
  • Operators: The arithmetic operators like the addition sign, subtraction sign, equal sign, etc., are known as operators.

An equation may have only numeric terms, algebraic terms, or both. We shall learn the steps to solve an equation in the next module.

How to Solve an Equation?

You can imagine an equation as a weighing scale. Both sides of an equation result in the same value, hence keeping the weighing scale in balance. It still holds whether we add or remove the same amount from both sides of an equation. The same remains true whether we multiply or divide the same integer into both sides of an equation.

Whatever operation you perform on the left side must also be implemented on the right side of an equation. This is done to maintain the balance of the equation. Let us take an example to understand this concept.

Consider an equation x + 3y = 2a + b. This is a balanced equation. Let us say that we want to add 20 to the left-hand side. The left-hand changes to x + 3y + 20. But now, the equation has been disturbed. We need to add 20 to the right side to balance it. Therefore the new equation will be x + 3y + 20 = 2a + b + 20.

Let us now learn how to solve an equation:

  • Step 1: Always keep the variables to one side of the ‘=’ symbol and constants to the other side of the ‘=’ symbol. Normally, the variables are kept to the left-hand side and constants on the right-hand side of the ‘=’ sign. 
  • Step 2: Do the necessary operations on the like terms like addition or subtraction. Do not operate like terms with unlike terms. You will be following the wrong approach if you do so.
  • Step 3: Simplify the equation and get the desired answer. 

Look at the example given below:

Example 1: Solve the equation a + 34b – 22 = b + 18

Solution: 

From step 1: a + 34b – b = 18 + 22

From step 2: a + 33b = 40

From step 3: This equation cannot be further simplified hence a + 33b = 40, is the solution of this equation. 

Example 2: Solve the equation 4x + 3 = x + 27

Solution: 

From step 1: 4x – x = 27 – 3

From step 2: 3x = 24

From step 3:  x = 24/3, therefore the value of x and the solution to this equation is 8.

Types of Equation

The equations are classified based on the degree of variables. A degree means the power assigned to any variable. Variables having a degree of 1 are known as linear variables. Similarly, the variables having a degree of two are known as quadratic variables, and a degree of three are known as cubic variables, and so on. The equations are classified as linear equations, quadratic equations, cubic equations, etc. Let us study them in detail.

Linear equations: The equations having all the variables of degree as one are known as linear equations. For example, x + 2y = 11, x + 5 = 0, etc. These are further classified as linear equations in one variable, linear equations in two variables, and so on. The standard form of a linear equation is kx + ly – m = 0, where x and y are variables and k, l, and m are constants. 

Quadratic equations: The equations having at least one variable of degree two are known as quadratic equations. For instance, x2 + a = 22, k2 – m2 – n = 0, etc. kx2 + lx + m = 0 is the standard form of a quadratic equation. 

Cubic equations: The equations having at least one variable having a degree of 3 are known as cubic equations. For example, x3 – x2 + 2 = 31y , k3 – m2 + n3 = 19, etc. The standard form to write a cubic equation is kx3 + lx2 + mx + n = 0. 

In all the above cases k, the coefficients of x, x2, and x3 can never be zero. If k = 0, then the terms containing the degree will be eradicated, making the equation invalid.

For instance if in the equation kx2 + lx + m = 0, if k = 0, then the equation becomes lx +m = 0, which is not a quadratic but a linear equation. 

Key Notes on Equations

  • The solution or root of the equation refers to the values of the variable that make an equation true.
  • When the same number is summed, subtracted, multiplied, or divided into both sides of an equation, the solution remains unaltered.
  • A straight line graphs a linear equation in one or two variables, while a parabola graphs a quadratic equation.

Frequently Asked Question?

1. How do you write a complete equation?

You can write a complete equation by adding the equal sign to the beginning of an equation and then writing all of the variables, numbers, or symbols that appear on either side of it.

For example, x + 2 = 5 In this example, “x” is on one side of the equal sign and “2” is on the other side.

2. How do you write a complete equation?

In order to write a complete equation, you need to know how to solve for one variable at a time.

Let’s say we have the equation:

x + y = 12

To solve for x, we need to isolate it by subtracting y from both sides of the equation:

x + y – y = 12 – y

x = 12 – y

Now, if we want to solve for y, then we just do the same thing: subtract x from both sides and then divide by 2.

3. How does one solve the equation?

Let’s start with a simple equation:

x² + 3x + 2 = 0

To solve this equation, we’re going to need to isolate the x term. We can do that by adding or subtracting multiples of the same number to both sides of the equation. We will make our life easier by adding 6 to both sides of the equation. This gives us:

x² + 3x + 2 = 0 + 6 (add 6)

x² + 3x + 8 = 6 (subtract 6)

We now have an equation where we can isolate our x term:

(x² + 3x) – 8 = 0 (subtract 8 from both sides)

We can now solve for x: divide both sides by (x² + 3x):

1/2(x² + 3x) – 8/2 = 0 => 1/2(1+3*(-8)) -8/2 => 1/2(-11) -8/2 => 1/2(-11) – (-8/2) => 1/2(-11 – 16) => 1/2(-17)=> x=-17

4. What are the steps in an equation?

 An equation is a mathematical statement that two things are equal. There are four steps to solve an equation:

1. Find the value of one side of the equation by itself. For example, if you have 3+2=5, then 5=3+2.

2. Solve the variable in terms of other variables in the equation by subtracting or dividing both sides of the equation by something. For example, if you have 3x+2=5, then 3x-5/3=0 or -5/3 x = 0, so x = -1/3 or 1/3.

3. Substitute your answer back into the original equation and solve for another variable in terms of all others; this will give you another solution to check against. For example, if x = 1/3 and y = -1/3 (from step 2), then 3(-1/3)+2(-1/3)=5(-1/3). If you add these together, you get -4/9 = 5(-1/9) and -4 + 9 = 5, so -4 + 9 = 5; therefore x may be 1/3 but not necessarily 1/9! So we’re done here!

5. What are the types of equations?

There are several different types of equations:

* Linear equations are equations that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

* Quadratic equations have the form ax^2 + bx + c = 0.

* Absolute value equations are of the form |x|.

* Rational expressions are of the form p/q, where p and q are polynomials with no common factors and q ≠ 0.

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