Easy101: Linear Equations in One Variable

What is a Linear Equation in One Variable?

A linear equation in one variable is an equation that can be written in the form:

[ ax + b = 0 ]

Where:

• ( a ) and ( b ) are constants, and
• ( x ) is the variable.

The solution to such an equation is the value of ( x ) that makes the equation true.

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Examples of Linear Equations in One Variable

1. ( 2x + 5 = 15 )
2. ( 4x - 7 = 9 )
3. ( -3x + 8 = 2 )

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Solving Linear Equations in One Variable

To solve a linear equation, follow these steps:

Step 1: Simplify the Equation

• Combine like terms on both sides of the equation.

Step 2: Isolate the Variable

• Use addition, subtraction, multiplication, or division to isolate ( x ) on one side of the equation.

Step 3: Solve

• Perform the necessary operations to find the value of ( x ).

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Worked Example

Solve the equation:
[ 3x + 7 = 16 ]

Step-by-Step Solution:

1. Simplify the Equation:
   Subtract ( 7 ) from both sides:
   [ 3x = 16 - 7 ]
   [ 3x = 9 ]

2. Isolate the Variable:
   Divide both sides by ( 3 ):
   [ 3x/3 = 9/3 ]
   [ x = 3 ]

Final Answer:

[ x = 3 ]

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Real-Life Applications of Linear Equations

Linear equations are used in everyday life to solve practical problems, such as:

1. Budgeting: Calculating expenses and savings.
   Example: If you spend ( $10 ) on food and want to save ( $50 ), how much must you earn?
   Solve ( x - 10 = 50 ).

2. Distance Problems: Calculating travel distance or time.
   Example: If a car travels (60 km/hr), how long does it take to travel (180 km)?
   Solve ( 60x = 180 ).

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Practice Problems

Try solving these linear equations:

1. ( 5x - 3 = 12 )
2. ( 7 + 2x = 17 )
3. ( -4x + 8 = 20 )

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Linear equations in one variable are the foundation of algebra and are essential for solving more complex math problems. Keep practicing, and you’ll master them in no time!

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