Linear Equations: You must already be aware of algebraic expressions and polynomials. The only difference between a linear equation and the 2 mentioned before is the equals to sign “=”. You can represent the equation as 2x – 3 = 7, where:
2x – 3 is LHS and 7 is RHS and = is equality sign.
Here we will provide you with all the information on linear equations such as definition, formula, sample questions and examples as well.
Linear Equations Definition
Any equation having the maximum order of 1 with both LHS and RHS terms separated by equality is called a linear equation.
Some examples are:
- a) 3x – 3 = 0,
- b) 4y = 8
- c) m +9 = 0,
- d) x/5 = 3
- e) x + y = 10
- f) 4x – y + z = 12
Types Of Linear Equations
Ther are 3 types of equations based on the variable type and theses have been provided in the table below:
|Linear Equation in One variable||Linear Equation in Two variable||Linear Equation in Three variable|
|4x + 8 = 0|
2x – 12 = 0
7x = 49
|y + 5x = 38|
a + 3b = 60
x + 2y – 12 = 0
|x + y + z = 12|
a – 3b = 2c
2x + 6y = ½ z
Solving Linear Equations
A single method is used to solve any types of linear equation irrespective of the number of variables. Lets us show you this with examples.
Example 1: Find the solution of 2x – 3 = 7.
- 1st Step: write the equation as it is 2x – 3 = 7.
- 2nd Step: Shift the constant part towards the right. It is to be noted that the sign before the constant will change when taken toward right i.e. + becomes – & * become ÷ and vice versa.
- 3rd Step: The equation will look like “2x = 7 + 3”.
- 4th Step: Perform mathematical operations. So, “2x = 10 => x = 10/2 => x = 5”.
The required solution for the above equation is x = 5.
Example 2: Find the solution of 10x = 30 – 5x.
- 1st Step: write the equation as it is 10x = 30 – 5x.
- 2nd Step: Bring the variables together i.e. 10x + 5x = 30.
- 3rd Step: The equation will look like “15x = 30”.
- 4th Step: Perform mathematical operations. So, “x = 30/15 => x = 2”.
Example 3: Solve x + y = 15 and 2x – y = 20
Give names to the equations
x + y = 15 as (a) and 2x – y = 20 as (b)
Find the value of x or y in equation (a)
x = 15 – y or y = 15 – x.
Now substitute the value of either x or y in equation (b)
- 2x – 15 – x = 20
- x – 15 = 20
- x = 20 + 15
- x = 35.
Now, substitute the value of x in equation (a) i.e. x + y = 15.
- 35 + y = 15
- y = 15 -35
- y = -20
The solution for the above equation is x = 35 and y = -20.
Linear Equations Formulas
Here are some formulas or forms with example that will help you understand the equations better.
|Linear Equation||General Form||Example|
|Slope intercept form||y = mx + c||y + 3x = 4|
|Point–slope form||y – y1 = m(x – x1 )||y – 2 = 2(x – 3)|
|General Form||Ax + By + C = 0||4x + 5y – 6 = 0|
|Intercept form||x/x0 + y/y0 = 1||x/3 + y/2 = 4|
|As a Function||f(x) instead of yf(x) = x + C||f(x) = x + 5|
|The Identity Function||f(x) = x||f(x) = 4x|
|Constant Functions||f(x) = C||f(x) = 12|
Here are some questions that you an solve and check whether your concepts are clear or not. These questions are also helpful from the examination point of view as well.
|Q1. Solve 2x – 3 = x + 2 |
A1. x = 5.
|Q2. Solve 5x + 7/2 = 3/2x – 14|
A2. x = -5
|Q3. Check your results for the below equations: |
1. 3x = 2x + 18
2. 5t – 3 = 3t – 5
3. 5x + 9 = 5 + 3x
4. 4z + 3 = 6 + 2z
5. 2x – 1 = 14 – x
6. 8x + 4 = 3 (x – 1) + 7
|Q4. The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. What can be the original number? |
|Q5. Arjun is twice as old as Shriya. Five years ago his age was three times Shriya’s age. Find their present ages. |
A5. Shriya’s present age = 10 years. Arjun’s present age = 20 years.
NCERT Solutions For Linear Equations
Students searching for the NCERT solutions on Linear Equations In One Variable and Pair Of Linear Equations In Two Variables chapters but end up on a website that either asks for registration or some amount to access the solutions. Here at embibe we provide these for free to the students. just click on the link and download the pdf from the page.
|NCERT Solutions For Class 8 Maths Chapter 2|
|NCERT Solutions For Class 10 Maths Chapter 3|
Frequently Asked Questions – FAQs
Check out the questions that are mostly searched on the topic.
|Q. What is a Linear Equation? Explain with an Example. |
Ans. Any equation having a variable, a constant and equality sign and can be represented as ax + by = c is called a linear equation. Example:
1. 5x – 7y = 4
2. 3x – 0y = 10 => 3x = 10.
|Q. What is the Formula for a Linear Equation? |
Ans. The formula or expression for the equation is ax + by = c. Where a, b, c are constants such as 1, 2, 35, 109, etc and x and y are variables.
|Q. What is the difference between linear and non-linear equations?|
Ans. A linear equation is valid for straight lines only whereas a non-linear equation can be a curve that has a variable slope value.
We hope you enjoyed learning about Linear Equations and the content answered all your queries. However if you have further questions feel free to use the comments section below and we will provide you with an update.561 Views