How to Solve Different Types of Equations Easily
What are Equations in Math?
Equations are something that we can understand as the combination of numbers, operators on the left hand side and the result on the right hand side separated by an equal to ‘=’ sign’. The operators that we study in math are +, -, , and .
For example, my mom brought two chocolates of Dairy milk and three 5-star chocolates for my younger sister, so how many chocolates does my sister have? Well, here, we will be using the addition (+) operator because the total is being asked here.
2 + 3 = 5
So, the numbers on adding give 5 as the result on the right-hand side. This is how we understand equations in math. Here, on this page, we will go through certain more equation examples.
Certain Examples for Equations in Math
Example 1: Yesterday, Shrishti purchased apparel worth Rs. 350. Today, she decided to buy another dress but before that she has to count how much she is left with. Like earlier, she had Rs. 1000 in her pocket, so which approach she follows to determine the amount she has at present?
Answer: Well, it is so easy to determine the remaining amount so that she could decide either to buy a new dress or not.
So, original amount she had = Rs. 1000
Amount she spent yesterday = Rs. 350
Now, the amount she has left with is ‘?’
So, 350 + ? = 1000
Here, when we take 350 on the right-hand side, the sign before it changes from ‘+’ to ‘-’, so we have:
? = 1000 - 350
Please note that the process of moving a term from one side of the equation to the other side and changing the sign before the term is transposition.
And, now using the concept of borrowing (subtraction regrouping), we have:
Here, 5 is greater than 0, so we borrow ‘1’ from the next place value, which is again ‘0’ and then again we go the other digit at the next place value, we find ‘1’ there. Now, when we take ‘1’ from here, the number in the minuend becomes 10 which is greater than ‘5’ in the subtrahend, so we get 10 - 5 = 5.
Also, earlier in one’s place, we had 0 - 0, which was clearly ‘0’.
So, last two digits becomes ‘50’, now moving to the next digit, which after giving ‘1’ to the minuend corresponding to ‘5’ in the subtrahend becomes ‘9’ in place of ‘10’, so now we have a new minuend corresponding to ‘3’ as 9 and subtracting 9 from 3, i.e., 9 - 3 = 6.
Now, our answer becomes 650. This means that Shrishti is left with Rs. 650 after purchasing apparel of Rs. 350 yesterday.
Example 2: Suppose your brother’s height is three times your height and this is just an assumption, means that you are not aware of your height. Now, if you subtract ‘5’ from your brother’s height and suppose that 13 is the result, how do you interpret this information in an equation form?
Answer: Well, it is pretty simple to understand and follow the underlines steps:
Step: Assume that your height is ‘b’ cm and then your brother’s height becomes three times, i.e., ‘3b’ cm. And here you have assumed that the result is around 13 cm after subtracting ‘5’ from your brother’s height, so let us form an equation now:
3b - 5 = 13
3b = 13 + 5 (‘-’ sign on the LHS becomes ‘+’ when migrating to the right-hand side)
3b = 18
b = 18 3
So, b = 6 cm
Hence, your interpretation for the height turns out to be a single digit number, i.e., your height is 6 cm.
From the above text, we understand that equations are statements of equality between two expressions which are composed of numbers, variables, and operators. Going through these examples will help you understand what equations are and how we solve them.
FAQs on Equations Examples: Learn with Step-by-Step Solutions
1. What is an equation in mathematics?
An equation is a mathematical statement that asserts the equality of two expressions. It is defined by the presence of an equals sign (=), which separates the statement into a left-hand side (LHS) and a right-hand side (RHS). For example, in the equation 4x + 5 = 13, '4x + 5' is the LHS and '13' is the RHS. The core purpose of an equation is to show a relationship of balance between the two sides.
2. Can you provide some simple examples of equations?
Certainly. Here are a few basic examples of equations that show different forms:
- Arithmetic Equation: 7 + 8 = 15. This shows equality between two numbers.
- Linear Equation in One Variable: 2y - 3 = 11. Here, 'y' is the unknown variable we can solve for.
- Equation with Variables on Both Sides: 5x + 2 = 3x + 10. The goal is to find the value of 'x' that makes both sides equal.
- Quadratic Equation: x² - 4 = 0. This is an equation of the second degree because the highest power of the variable is 2.
3. What is the fundamental difference between an equation and an expression?
The fundamental difference lies in the presence of an equals sign (=). An equation contains an equals sign and states that two expressions are equal, like `3x + 7 = 16`. It can be solved. An expression, on the other hand, is a combination of numbers, variables, and operators without an equals sign, such as `3x + 7`. It represents a value that can be simplified or evaluated, but it cannot be solved as there is no relationship of equality stated.
4. What are the main types of equations students learn as per the CBSE syllabus?
According to the CBSE curriculum, students progressively learn several types of equations. The main ones include:
- Linear Equations in One Variable: These are the simplest algebraic equations, like `ax + b = 0`, typically introduced in middle school (Classes 6-8).
- Linear Equations in Two Variables: These are of the form `ax + by + c = 0` and are studied as pairs of simultaneous equations in Classes 9 and 10.
- Quadratic Equations: These are second-degree equations of the form `ax² + bx + c = 0`, a key topic in Class 10.
- Cubic Equations: While not as central as quadratics, these third-degree equations are also introduced in higher classes.
5. Why is the 'equals' sign so important in an equation?
The equals sign is crucial because it signifies balance and relationship. It turns a simple expression into a statement of equality, indicating that the value on the left-hand side is exactly the same as the value on the right-hand side. This principle of balance is what allows us to solve the equation. Any operation we perform on one side (like adding, subtracting, multiplying, or dividing) must also be performed on the other side to maintain this balance and find the value of the unknown variable.
6. How are equations used as examples to solve real-world problems?
Equations are powerful tools for translating real-world scenarios into a mathematical format that can be solved. For example:
- Budgeting: To figure out how many notebooks ('n') you can buy at ₹25 each with a total of ₹150, you can set up the equation `25n = 150`.
- Calculating Distances: The famous equation `Distance = Speed × Time` is used to plan journeys. If you travel at 60 km/h for 3 hours, the equation `D = 60 × 3` tells you the distance covered.
- Temperature Conversion: The equation `F = (9/5)C + 32` is used to convert temperatures from Celsius (C) to Fahrenheit (F).
7. What would be an example of something that is NOT an equation?
Anything that is a mathematical statement without an equals sign is not an equation. A common non-example is an expression. For instance, `7x - 5` is an algebraic expression, not an equation. Another important non-example is an inequality. A statement like `x > 10` uses a 'greater than' symbol, not an equals sign. It shows a relationship where one side is larger than the other, rather than equal, so it is not an equation.
