How to Apply Equations and Formulas in Maths Problems
What is an Equation?
Equations are vital parts of our lives. We use equations not only in subjects like Mathematics and Science but also to calculate the price, debt, tax, interest, etc. The best example of an equation is “ 5 + 5 = 10”. But exactly, what is an equation?
An equation is an analytical statement that states that two things are equal. An equation includes either term or expression. In Mathematics, an equation is defined as equality including one or more variables. Solving equations means determining which values of the variables make the equation true. In this case, variables are considered as unknown and the values which satisfy the equality are known as solutions. An equation differs from identity in that equation is not certainly valid for all possible values of the variables.
The (“=”) symbol which can be seen in every equation, was introduced by Robert Recorde in 1557, who stated that nothing can be more equal than the parallel straight lines with similar length.
Define Equation
An equation is a Mathematical statement with an ‘equal to’ sign between two algebraic expressions that have similar values.
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For example, 5x + 9 is the expression on the left-hand side which equals to the expression 24 on the right-hand side.
Look at the following examples of equations. This will give you a clear idea of what the equation is in Maths.
Equation Examples
What is Formula?
In Mathematics, a formula is a fact or rule expressed with a Mathematical symbol.
The formula generally includes:
An ‘ equal to’ symbol.
Two or more variables (x, y, etc)
For Example,
The perimeter of a rectangle formula is
Perimeter = 2 (Length + Breadth)
if the length and width of the rectangle are ‘a’ and ‘b’ respectively, the formula of its perimeter is
Perimeter = 2 ( a + b)
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Formula Without “ Equal to” Symbol
Sometimes, the formula can be expressed without ‘ equal to” sign
For Example,
But this can also be expressed with an ‘ equal to’ symbol as it can also be written as Perimeter of a square = 4 x side.
What is the Subject of a Formula?
The subject of a formula is the single variable which is expressed in terms of other variables included in the formula.
Formulas are written so that a single variable that is the subject of the formula is written on the left-hand side of the equation and everything else goes on the right side of the equation.
For Example,
In the formula v = u + at, v is considered as the subject of the formula.
How to Change the Subject of a Formula?
To change the subject of a formula, items in the formula need to be rearranged, so that new variables become the subject of the formula. Knowledge of solving equations and inverse operations is important to have while changing the subject of a formula.
For example,
In formula A= bh, area (A) is the subject of the formula which means it is the area that has to be calculated.
If the area and height of a rectangle are given and you are asked to calculate the base of a rectangle, the formula A= bh will not be helpful to calculate the area of the rectangle as now ‘b’ has to be calculated.
To calculate b that is the base of a rectangle, the formula A = bh has to be reordered to make bas the subject of the formula.
To make bas the subject of the formula, b needs to be isolated. In the above formula, the variable b is multiplied by the variable h. The inverse of multiplying the variable by h , is dividing it by h.
In the formula A = bh we will divide both the side by h to isolatebas shown below:
A/h = bh/h
A/h = b
The variable b is now the subject of a formula.
Now, we will use the formula A/h = b, to calculate the base of a rectangle.
Solved Examples
1. Rearrange the volume of a vox formula (V = lwh) to make w that is the width as the subject of a formula.
Solution:
We will start with,
V = lwh
Dividing both the sides by h
V/h = lwh/h
We get,
V/h = lw
Dividing both the sides by l
V/hl = lw/l
We get,
V/hl = w
Swapping the sides, we get
w = V/hl
Now, you can easily calculate the width of a box by applying the formula w= Vhl, where w is the width, v is the volume, and h is the height of the box.
2. Find the value of a, If 5a + 9a = 16 - 2a
Solution:
We have,
5a + 9a = 16 - 2a
5a + 9a + 2a = 16
16 a = 16
a = 16/16
a = 1
Fun Facts
The first formula was introduced in between 1800 - 1600 BC.
The first equation was written by Robert Recorde in his treatise “ The Whetstone of Witte” in 1557. In modern terms, the equation is represented by 14x + 15 = 71, and its solution is x = 4.
FAQs on Equations and Formulas: Definitions, Types, and Uses
1. What is the basic definition of an equation and a formula in Maths?
In mathematics, an equation is a statement that asserts the equality of two expressions. It always contains an equals sign (=) and is used to find the value of an unknown variable. A formula, on the other hand, is a specific type of equation that expresses a rule or relationship between two or more variables. It is a set of instructions for calculating a desired quantity.
2. What is the main difference between an equation and a formula?
The primary difference lies in their purpose. An equation's main purpose is to state a condition of equality to be solved for an unknown value (e.g., solving 2x + 5 = 15 to find x). In contrast, a formula's main purpose is to provide a computational rule that can be used repeatedly to find a result, given the values of its variables (e.g., using A = πr² to find the area of any circle).
3. Are all formulas equations? And are all equations considered formulas?
This is a common point of confusion. Yes, all formulas are equations because they use an equals sign to show a relationship between expressions. For example, the formula for the perimeter of a rectangle, P = 2(l + w), is an equation. However, not all equations are formulas. An equation like x + 7 = 10 is simply a statement to be solved for a specific value of 'x'. It doesn't represent a general rule or principle, so it is not considered a formula.
4. What are the common types of equations studied in the CBSE syllabus?
As per the CBSE curriculum for various classes, students primarily study the following types of equations:
- Linear Equations: Equations where the highest power of the variable is 1. They can be in one variable (e.g., 3x - 6 = 0) or two variables (e.g., y = 2x + 1).
- Quadratic Equations: Equations where the highest power of the variable is 2, typically in the form ax² + bx + c = 0.
- Polynomial Equations: A general category where an equation is formed by a polynomial set to zero. This includes linear, quadratic, and cubic (highest power is 3) equations.
- Simultaneous Equations: A set of two or more equations with the same variables that are solved together to find a unique solution.
5. Can you provide some examples of mathematical equations and formulas?
Certainly. Here are some clear examples:
- Example of an Equation: 5y - 10 = 25. Here, the goal is to find the specific value of 'y' that makes the statement true.
- Example of a Formula: The Pythagorean theorem, a² + b² = c². This is a formula used to find the length of a side of any right-angled triangle when the other two sides are known.
- Another Formula Example: The formula for simple interest, I = P × R × T, which calculates the interest (I) based on the principal (P), rate (R), and time (T).
6. Why is it important to learn how to form and solve equations?
Learning to form and solve equations is a fundamental skill because it teaches logical reasoning and problem-solving. It allows us to model real-world situations mathematically. For instance, equations are used to calculate profits in business, determine trajectories in physics, balance chemical reactions, and manage personal finances. It provides a powerful tool to find unknown quantities and make informed decisions based on known relationships.
7. How does a simple mathematical statement become an equation?
A mathematical statement becomes an equation at the exact moment two expressions are set equal to one another using the equals sign (=). An expression on its own, like 'x + 5' or '12', is just a value or a representation of a value. It only becomes an equation when it is part of a statement of balance, such as 'x + 5 = 12'. The equals sign is the critical component that transforms expressions into a relationship that can be solved or analysed.
8. What are the essential parts of a mathematical equation?
Every mathematical equation is composed of several key parts:
- Variables: Symbols (like x, y, a) that represent unknown or changing quantities.
- Constants: Fixed numerical values that do not change (e.g., 5, -10, π).
- Coefficients: A number multiplied by a variable (e.g., the '3' in 3x).
- Operators: Symbols that indicate an operation, such as addition (+), subtraction (-), multiplication (×), and division (÷).
- The Equals Sign (=): The core of the equation, indicating that the expression on the left side has the same value as the expression on the right side.
