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Last updated on September 15, 2025
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.
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.
Although both expressions and equations are used in algebra, they differ 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 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 |
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 student's ability to solve problems related to this topic. Given below is a list and explanation of different types of equations:
Linear equations are those equations where the highest degree of variables is 1. They can be classified based on the number of variables. Example: 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.
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:
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
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)
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:
Take a look at the expression given below:
(3 × x) + 15 = 24
Here,
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. When moving the constant -2 to the RHS, its sign changes to +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
Equations are not just used to solve mathematical problems. They are used to solving practical problems as well. Some real-life applications are:
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.
Solve 3(x + 4) - 6 = 18
4
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
Solve 6y + 8 = 3y
The value of y is 2
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 = -2 + 8
3y = 6 ⇒ y = 2
Therefore, the value of y is 2
Solve x² + 6 = 31
The result is ±5
x2 + 6 = 31
x2 = 31 - 6
x2 = 25
x = √25 = ±5
Find the value of y in the equation (y + 3) / 2 = (y - 1) / 4
The value of y is -7
Given equation: (y + 3) / 2 = (y - 1) / 4
To remove the fraction, we can multiply the equation by the LCM of 2 and 4.
LCM of 2 and 4 = 4
Multiplying both side by 4:
4 × [(y + 3) / 2] = 4 × [(y - 1) / 4]
⇒ 2y + 6 = y - 1
⇒ 2y - y = (-1) + (-6)
⇒ y = -7
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 LHS and RHS equal 25
By solving the LHS and RHS separately, we can determine if they have the same value.
5 (x - 2) = 5 (7 - 2) = 5 × 5 = 25
The LHS is 25
3x + 4 = (3 × 7) + 4 = 21 + 4 = 25
The RHS is 25
Therefore, LHS = RHS
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.
: He loves to play the quiz with kids through algebra to make kids love it.