Solving Linear Equations: Class 10 Maths

Study Notes

Solving Linear Equations in Class 10 Maths

Linear equations play a crucial role in the field of mathematics, particularly during the course of study in high school algebra. They help us understand various mathematical concepts and their applications. In this article, we will discuss how to solve linear equations, which is a fundamental concept taught in class 10 maths.

Understanding Linear Equations

A linear equation is defined by an expression in one variable where the highest power of the variable is one. For example, consider the following expressions: a) 2x + 3 = 5 b) y - 4 = x/3 c) z = 7 - t

The first two expressions are examples of linear equations with different variables (e.g., x or y), while the third one is a linear equation in terms of two variables (z and t).

Steps for Solving Linear Equations

To solve a linear equation, follow these steps:

Step 1: Simplify both sides of the equation

You can simplify both sides using addition or subtraction rules. This step helps you isolate the value of the unknown variable (x, y, or z) on one side by getting rid of its exponents, if any, or by moving it to the left or right side depending on the sign used in this step.

Step 2: Get rid of fractions by multiplying both sides by their LCM

If there are any fractions present, you need to get rid of them using multiplication rules, applying the same operation for both sides and ensuring that the unknown variable remains unchanged.

Step 3: Solving the equation for the desired value

Once the equation has been simplified and all fractions have been eliminated, you can solve it easily. Depending on whether you want to find x, y, or z, you will either move that variable to one side of the equal sign or write the equation in terms of the other variables.

Here's an example problem to illustrate these steps:

Solve the following equation for x:
x + 5 = 9

Step 1: Subtract 5 from both sides of the equation to simplify it:
x + 5 - 5 = 9 - 5
x = 4

Solving Linear Equations with Two Variables

Solving linear equations with two variables (e.g., z and t) involves using substitution or elimination methods. Let's discuss these methods one by one.

Substitution Method

The substitution method allows you to solve linear equations involving two variables by first solving for one of them, say z, and then plugging this value back into the equation that contains both variables, allowing you to find the other variable, say t.

For example, consider the following system of equations:

z + t = 10 z = x + 3

By solving the second equation for z, we get:

z = x + 3

Now, substitute z from the first equation into the second equation:

(x + 3) + t = 10 t = 10 - x - 3 t = 7 - x

This shows that the solutions for the linear equations with two variables can be found through substitution.

Elimination Method

The elimination method entails adding or subtracting certain combinations of the given equations to eliminate one of the variables, thus making it easier to solve for the remaining variable.

For instance, consider the system of equations:

z + t = 10 z = 3x - 5

In order to eliminate z, we add both equations:

((3x - 5) + (z)) + t = 10 3x - 5 + z + t = 10

Now, we can substitute z from the first equation into the second one:

3x - 5 + (3x + 3) + t = 10 t = 10 - 3x - 3 t = 7 - 3x

Thus, we have found the solutions for the linear equations with two variables using the elimination method.

In conclusion, solving linear equations in class 10 maths is an essential skill that helps students understand the concept of algebra and its applications. By following the steps outlined in this article and practicing with various problems, students can become proficient in solving linear equations and continue to develop their mathematical abilities.

Description

Learn to solve linear equations in class 10 mathematics, a fundamental concept in algebra. Understand the steps involved in solving linear equations with one variable and with two variables through substitution and elimination methods. Practice problems to enhance your skills in solving linear equations.