Short Answer

To solve a linear equation in one variable, we find the value of the unknown variable by using basic operations like addition, subtraction, multiplication, and division. The goal is to make the variable alone on one side of the equation.

We do this by moving constants to the other side and simplifying step by step. At the end, we get the value of the variable that makes the equation true.

Detailed Explanation:

Solving a linear equation in one variable

Solving a linear equation in one variable means finding the exact value of the unknown variable that makes the equation correct. A linear equation has only one variable, and its highest power is 1. The general form of such an equation is ax + b = 0.

The main idea behind solving a linear equation is to isolate the variable on one side of the equation. This means we try to keep the variable alone on one side and all numbers (constants) on the other side. To do this, we use basic algebraic operations such as addition, subtraction, multiplication, and division.

The important rule to remember is that whatever operation we perform on one side of the equation, we must perform the same operation on the other side. This keeps the equation balanced.

Steps to solve the equation

There are simple steps that help in solving any linear equation easily.

First, remove brackets if there are any by multiplying terms inside the brackets.

Next, combine like terms on both sides of the equation. This means adding or subtracting similar terms.

Then, move all terms with the variable to one side of the equation and constants to the other side. This is done by adding or subtracting terms from both sides.

After that, simplify the equation as much as possible.

Finally, divide or multiply to make the variable alone. This gives the final value of the variable.

These steps help in solving even slightly complex linear equations in a clear and organized way.

Example of solving equation

Let us take a simple example:

x + 5 = 12

To solve this, we subtract 5 from both sides:

x = 12 − 5
x = 7

So, the value of x is 7.

Now let us take another example:

2x + 4 = 10

First, subtract 4 from both sides:

2x = 10 − 4
2x = 6

Now divide both sides by 2:

x = 6 ÷ 2
x = 3

So, the value of x is 3.

Checking the solution

After solving an equation, it is always good to check the answer. To check, we put the value of the variable back into the original equation.

For example, if x = 3 in the equation 2x + 4 = 10:

2(3) + 4 = 6 + 4 = 10

Since both sides are equal, the solution is correct.

Checking helps us avoid mistakes and confirm that our answer is right.

Importance of solving equations

Solving linear equations is very important in mathematics. It helps us find unknown values easily and understand relationships between numbers.

In daily life, we use these equations in many situations. For example, if you know the total cost and the price of one item, you can use an equation to find how many items you bought.

They are also used in calculating time, distance, and speed. In business, they help in finding profit and loss. In science, they are used to solve simple formulas.

Learning how to solve equations improves logical thinking and problem-solving skills.

Common mistakes to avoid

While solving linear equations, some common mistakes should be avoided.

Do not forget to perform the same operation on both sides of the equation.

Be careful with signs, especially when subtracting or adding negative numbers.

Do not skip steps, as it can lead to incorrect answers.

Always simplify the equation properly before finding the final answer.

Avoiding these mistakes helps in getting correct results.

Conclusion

To solve a linear equation in one variable, we isolate the variable by using basic algebraic operations step by step. This process helps us find the unknown value easily. It is an important skill used in mathematics and real-life problem solving.