106 LearnersLast updated on July 4th, 2025
Equation
In mathematics, when we need to find the value of a variable, we can use an equation. This is called a mathematical equation, and it consists of two sides, the left-hand side (LHS) and right-hand side (RHS). An equal sign separates the LHS and the RHS of any equation.
What are Equations?
Equations show how LHS and RHS are related. We use them to find unknown variables. Without an = sign, we cannot write an equation.
For example, a = 10b - 5 is an equation, whereas p3+q - 6 is not an equation.
Difference Between Expression & Equation
Even though we use both expressions and equations in algebra, they are different from each other. The table below illustrates the differences between them.
| Expression | Equation |
| An expression can be written without an equal to sign. | An equation cannot be written without an equal to sign. |
| A mathematical expression consists of one or more terms connected by operations, such as addition, subtraction, etc. | In an equation, two expressions are equal. These two expressions are represented using the = sign. |
| Example: x - y + 8 | Example: 2b + 4 = c + 6 |
What are the Types of Equations?
Equations are categorized based on their degree, which is the highest power of the variable in the given equation. Learning about different types of equations will improve students ability to solve problems related to this topic. Given below is a list and explanation of different types of equations:
- Linear Equation
- Quadratic Equation
- Cubic Equation
- Rational Equation
Linear Equation
Linear equations are those equations where the highest degree of variables is 1. They can be classified based on the number of variables. E.g., equations with only one variable are called one-variable linear equations, and so on. The general form of a linear equation can be represented as:
aP + bQ + c = 0, where a and b are the coefficients of the variables P and Q, while c is the constant. Here, the highest degree of variables P and Q is 1.
Quadratic Equation
In quadratic equations, the highest degree of the variable will always be 2. The general form of a quadratic equation with m as the variable will be:
am2 + bm + c = 0 where a ≠ 0.
In the equation:
- m2 is the variable with 2 as the highest degree, and a is the coefficient of m2. If a is zero, then the equation cannot be quadratic.
- c is the constant.
Cubic Equation
In cubic equations, one of the variables will have 3 as the highest degree. The general form of a cubic equation will be:
ax3 + bx2 + cx + d = 0, where x3 is the variable with the highest degree. a, b, and c are coefficients of x3, x2, and x respectively, and d is the constant.
Rational Equation
Equations containing a fraction, where the numerator or denominator or both contain a variable, are rational equations. The rational equation is:
b/3 = a+c/4
What are the Parts of an Equation?
We know that an equation comprises LHS and RHS connected by the equal to sign. Apart from the LHS and RHS, an equation also includes the following components:
- Coefficient: A number that multiplies a variable
- Variable: A letter or symbol that represents a value
- Operator: Symbols like +, -, ×, to show different arithmetic operations
- Constant: A value that doesn't have any variable attached to it
Take a look at the expression given below:
3x + 15 = 24
Here, the LHS is 3x + 15 and RHS is 24
In 3x, 3 is the coefficient of the variable x
+ is the operator
15 and 24 are the constants
How to Solve Equations?
An equation means LHS should be equal to RHS. Any changes made on the LHS should be done on the RHS too. To find an unknown variable, we must solve the equation while maintaining the balance on both LHS and RHS.
Let's see how to solve a basic equation, 3x - 2 = 4. Follow the steps given below:
Step 1: Make the terms with variables to be on one side and the constants on the other side. Here, 3x will remain on the LHS, and 2 will go to the RHS. Since the constant -2 is going to the RHS, it will become +2.
Step 2: Now the equation will be 3x = 4 + 2
Step 3: Add the values on the RHS. The equation will be 3x = 6
Step 4: To find the value of x, we need to shift 3 to the RHS. Since 3 and x are connected by multiplication on the LHS, shifting 3 to the RHS will change its sign from multiplication to division. So, dividing 6 by 3, we get the value of x as 2
→ 3x = 6 → x = 6/3 = 2
Real-Life Applications of Equations
Equations are not just used to solve mathematical problems. They are used to solving practical problems as well. Some real-life applications are:
- Budget and Finance: Equations can track your expenditure and tell you how much you can save in a month.
- Time and Speed: To calculate the time to reach the destination or the speed required to get there, use the formula Speed = Distance/Time
- Electricity Usage: Used to calculate the electricity consumption of any device using the formula: Energy (in kWh) = Power (in kW) × Time (in hours)
Common Mistakes and How to Avoid Them in Equations
Students can make mistakes while solving equations, resulting in wrong answers. The mistakes can happen in any equation, such as linear, quadratic, cubic, or rational. Given below are some mistakes that can happen while solving equations. To overcome these, solutions have been provided.
Mistake 1
Forgetting to change the sign of the constant.
Children often forget to change the sign of the constant when moved to another side.
For example, the equation 3x - 6 = 9 can be wrongly solved as 3x = 9 - 6. We find the value of x as 1, which is incorrect. The correct way to solve the equation is:
3x = 9 + 6
3x = 15
x = 15/3 = 5
Here, -6 changed to +6 when moved to the RHS.
Always change the sign of the constant when moved to the other side.
Mistake 2
Incorrect grouping of terms
Always remember to group all constants and terms with the same variables. Children might mistakenly add the terms with variables directly to the constants, leading to incorrect results.
For example, in the equation 2x + 5 = 10 - 3x, we need to bring the terms with the variable to one side and the constants to the other side. Hence, the equation will be formed as
2x + 3x = 10 - 5
Solving the equation, we find the value of x as 1.
Mistake 3
Skipping steps while solving the equation
Skipping steps while solving results in incorrect answers.
For example, in equations like 2(x + 4) + 6x - 8 = 16, children can forget to expand the terms given in brackets. The correct way to solve the equation will be as follows:
2(x + 4) = 2x + 8
Therefore, 2(x + 4) + 6x - 8 = 16 can be solved as
2x + 8 + 6x -8 = 16
2x + 6x = 16
8x = 16
x = 16/8 = 2
Mistake 4
Forgetting to check if the equation is solved correctly using the final result
To see if you have solved the equation correctly, always substitute the value of the variable in the equation and see if LHS equals RHS.
For example,
Let x be 4
Substitute the value of x in the equation 2 (x + 3) - 4 = 10.
2 (x + 3) = 2 (4+3) = 2 x 7 = 14
2 (x + 3) - 4 = 14 - 4 = 10
Since LHS equals RHS, we can say that the equation is solved correctly
Mistake 5
Ignoring the ± sign while solving quadratic equations
While solving any quadratic equation, you need to remember that the square root of a variable will always be ±.
For example, in the equation:
x2 = 9
x = √9 = ±3
Here, x can either be +3 or -3
Solved Examples of Equations
Problem 1
Solve 3(x + 4) - 6 = 18
4
Explanation
First, expand the bracket 3(x + 4) → 3x + 12
After the expansion, the equation will be 3x + 12 - 6 = 18
Solving the equation, we get:
3x + 12 - 6 = 18
3x + 6 = 18
3x = 18 - 6
3x = 12
x = 12/3 = 4
Therefore, the value of x is 4
Problem 2
Solve 6y + 8 = 3y
The value of y is 2
Explanation
The 3y on the RHS should be moved to the LHS, and -8 must be moved to the RHS. By doing so, we can group the terms with y on the LHS and the constants on the RHS. Now the equation will look like this: 6y - 3y = -2 + 8.
Solving for y we get:
6y - 3y = 6
y = 6/3 = 2
Problem 3
Solve x² + 6 = 31
The result is ±5
Explanation
x2 + 6 = 31
x2 = 31 - 6
x2 = 25
x = √25 = ±5
Problem 4
Find the value of y in the equation y + 3/2 = y - 1/4
The value of y is -7
Explanation
Since the given equation is a fraction, we need to remove the fraction. To remove it, cross-multiply both sides by the denominators 2 and 4. Therefore, the equation will be 4 (y + 3) = 2 (y - 1). To find the value, expand the brackets and bring the variable to one side and the constants to the other side.
4 (y + 3) = 2 (y - 1)
4y + 12 = 2y - 2
4y - 2y = -2 - 12
2y = -14
y = -14/2 = -7
Problem 5
If x is given as 7, substitute 7 in the equation 5 (x - 2) = 3x + 4 and check if the LHS and RHS are the same.
Yes, the LHS and RHS are the same. Both sides give 25 as the answer.
Explanation
By solving the LHS and RHS separately, we can determine if they have the same value.
5 (x - 2) = 5 (7 - 2) = 5 x 5 = 25
The LHS is 25
3x + 4 = (3x7) + 4 = 21 + 4 = 25
The RHS is 25
Therefore, LHS = RHS
FAQs on Equations
1.Write the types of equations based on their degree.
2.How to solve an equation?
3.What do you mean by differential equations?
4.What is the main difference between an expression and an equation?
5.What is the general form of a quadratic equation?
6.How does learning Algebra help students in Vietnam make better decisions in daily life?
7.How can cultural or local activities in Vietnam support learning Algebra topics such as Equation?
8.How do technology and digital tools in Vietnam support learning Algebra and Equation?
9.Does learning Algebra support future career opportunities for students in Vietnam?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
