 × ## Introduction

An equation in mathematics is made up of two expressions separated by the equals symbol "=". The value on one side of an expression that is denoted by an equal sign must match the value on the other side. Equations containing variables on both sides might be difficult to solve, but with the appropriate strategy, you can identify the variable and determine its value. Since in these sorts of equations the variable appears on both sides of the equal sign, our objective is to move all the variable terms to one side of the equation and all the constant terms to the other. The usual method for doing this is to add or subtract terms from both sides until the variable terms are segregated. Once the variable is on one side of the equation, we may use basic algebraic operations to find its value.

## Solving equations having the variable on both sides

When an equation is solved with a variable on both sides, it signifies that the equation has two identical variables, one on the left side and the other on the right. Take the equation 3x + 2 = 3 - 2x, for instance. We have variables on both sides of this equation along with a few constants. There are several ways to solve these equations, including addition and subtraction. Therefore, x=2
Here by solving the following equation we get the value of the variable i.e  x=2.

### Solve for x equations

To "solve for x" is to determine the value of x in an equation where x is the only variable. Let's say we have been asked to identify the value of the variable in equation 7x - 14 = 0. If the equation's provided variable is x, and we have been asked to solve for x. Then we have to find the value of x

Let us take an example:

### Solved Problems

1. Solve for x in the given equation 4x + 5 = 2x - 3

Solution:

Step 1: Simplify both sides of the equation by combining like terms. To do this, we need to move all the x terms to one side of the equation and all the constant terms to the other side.

4x + 5 = 2x - 3

Subtract 2x from both sides:

4x - 2x + 5 = -3

Simplify:

2x + 5 = -3

Step 2: Isolate the variable term (2x) by moving the constant term (5) to the other side of the equation. To do this, we need to subtract 5 from both sides:

2x + 5 - 5 = -3 - 5

Simplify:

2x = -8

Step 3: Solve for x by dividing both sides of the equation by the coefficient of x (2):

2x/2 = -8/2

Simplify:

x = -4

So the solution to the equation 4x + 5 = 2x - 3 is x = -4.

2. The sum of the two numbers is 29. One of the numbers exceeds the other by 7. Find the numbers.

Solution:

Let the first number be x

Then the other number becomes = 7+x (according to the question)

Now,

The Sum of the two numbers is = 29

According to the question = x + x + 7 = 29

2x+7 = 29

2x = 29-7

2x = 22

∴ x = 11

Therefore, the First number is 11

The second number becomes = x + 7 = 11 + 7 = 18

Therefore, the two numbers are 11 and 18.

3. Solve the given equation: 3x + 2 = 17 Therefore, x=5
Here by solving the following equation we get the value of the variable i.e  x=5.

## Summary

Only one or two variables are present in a linear equation. A variable cannot be the denominator of a fraction or have a power larger than 1 in a linear equation. When you identify variable pair values that satisfy a linear equation and make it true. If a variable is present on both sides of an equation, we must combine like terms to simplify both sides, move all the variable terms to one side and all the constant terms to the other, and then use inverse operations to isolate the variable. By multiplying both sides by the variable's coefficient, we may finally determine the variable. By inserting our answer back into the original equation and verifying that all sides are equal, we may verify our conclusion.

Q1. What if there is no solution to the equation or an infinite number of solutions?

If an equation has no solutions, then no matter what value the variable has, the two sides of the equation can never be equal. Any value of the variable will make the two sides of the equation equal if the problem has an infinite number of solutions. In either scenario, you may find this by modifying the equation and simplifying it until it becomes obvious that there is either no solution or an infinite number of solutions.

Q2. What happens if the simplified variable terms don't cancel out?

You will need to segregate the variable term on one side of the equation if the variable terms don't cancel out after being simplified. To do this, you can add or subtract terms from both sides of the equation until just the variable term is left on one side and all the constants are on the other. To find the variable, divide both sides of the equation by the variable's coefficient.

Q3. Can exponents be used in linear equations?

Linear equations cannot use exponents other than 1. Ax + By + C = 0, where A, B, and C are any real integers and x and y are variables, which is how an equation is stated in its general form.

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