Linear Equations in One Variable

 

1. Linear Equation in One Variable Definition

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where x is the variable, and a and b are numbers. The term "linear" means that the variable, x, is raised to the power of 1, and there are no other powers or operations, like squares or cubes, on x. These equations are called "in one variable" because they involve only one unknown number.

 

For example:

1.    3x + 5 = 0

2.    2x - 7 = 13

3.    x + 9 = -4

 

2. Standard Form of Linear Equations in One Variable

 

The standard form of a linear equation in one variable is:

 

ax + b = 0

 

Here:

x is the variable we need to solve for.

a is the coefficient of x (it cannot be zero).

b is the constant term.

 

For example:

In   4x - 7 = 0, a = 4 and b = -7.

In  -3x + 2 = 0, a = -3 and b = 2.

 

 3. Solving Linear Equations in One Variable

To solve a linear equation, the goal is to find the value of x that makes the equation true. Here are the basic steps:

1. Simplify the equation: Combine like terms if needed.

2. Isolate the variable: Move all terms with x to one side of the equation and constants to the other side.

3. Solve for the variable: Divide or multiply to find the value of x.

 

 Example 1:

Solve 3x + 5 = 11.

 

Step 1: Subtract 5 from both sides.

            3x + 5 - 5 = 11 - 5

                       3x = 6

Step 2: Divide both sides by 3.

             3x/2 = 6/2         

                  x = 2

 

 Example 2:

Solve 7x - 3 = 25.

 

Step 1: Add 3 to both sides.

            7x - 3 + 3 = 25 + 3

                        7x = 28

Step 2: Divide both sides by 7.

                    7x/7 = 28/7

                         x = 4

 

4. Linear Equation in One Variable Examples

Here are more examples of linear equations and their solutions:

 

1. Solve 2x + 6 = 3x - 5

Answer:

2x + 6 = 3x - 5

2x + 6 - 6 = 3x - 5 - 6      (Subtract 6 to both sides)

          2x  = 3x - 11

    2x - 3x = 3x - 3x  - 11  (Subtract 3x to both sides)

           -x  = -11     

 -x  × (-1) = -11 × (-1)      (Multiply (-1) to both sides)

             x = 11                

 

2. Solve 5x - 12 = 2x + 3

Answer:

5x - 12 = 2x + 3

5x - 12 + 12 = 2x + 3 + 12      (Add 12 to both sides)

          5x  = 2x + 15

    5x - 2x = 2x - 2x  + 15        (Subtract 2x to both sides)

           3x  = 15     

       3x : 3 = 15 : 3                  (Devide 3 to both sides)

              x = 5                

 

3. Solve 3(2x - 5) = 4x + 7

Answer:

3(2x - 5) = 4x + 7

6x - 15 = 4x + 7

6x - 15 + 15 = 4x + 7 + 15      (Add 15 to both sides)

          6x  = 4x + 22

    6x - 4x = 4x - 4x + 22        (Subtract 4x to both sides)

           2x  = 22     

       2x : 2 = 22 : 2                  (Devide 2 to both sides)

              x = 11                

 

5. Linear Equation in One Variable Word Problems

Linear equations in one variable are useful for solving real-life problems. Let’s look at some examples:

 

Example 1: Age Problem

Sarah’s age is 5 years more than twice her brother’s age. If her brother is x years old and Sarah is 17 years old, find her brother’s age.

Answer:

The Equation is

2x + 5 = 17

Solution:

- Subtract 5 from both sides: 2x = 12

- Divide by 2: x = 6

Her brother is 6 years old.

 

 Example 2: Money Problem

A pencil costs $2 more than an eraser. If the pencil costs $10, find the cost of the eraser.

Answer:

The equation is

E + 2 = 10 (where E is the cost of the eraser)

Solution:

- Subtract 2 from both sides: E = 8

The eraser costs $8.

 

 Example 3: Distance Problem

A car travels at 60 km/h and covers a certain distance in t hours. If the total distance covered is 180 km, find the value of t.

Answer:

The equation is:

60t = 180

Solution:

- Divide both sides by 60 : t = 3

The car traveled for 3 hours.

 

Thank.