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A linear equation in one variable is a mathematical statement of equality involving a single unknown quantity (represented by a variable like 'x' or 'y'), where the highest power of that variable is 1. This means that the variable doesn't have exponents greater than 1, like x² or x³.

Standard form

The standard form of a linear equation in one variable is ax + b = 0, where:
'x' is the variable (the unknown).
'a' is the coefficient of the variable (a real number, and a ≠ 0).
'b' is a constant term (a real number).

For example, 5x + 3 = 0 and 2x - 1 = 5 are linear equations in one variable.

Solving linear equations in one variable

To solve a linear equation in one variable means finding the value of the variable that makes the equation true. Here's a general approach:
Clear fractions or decimals (if any): Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate fractions, or by a power of 10 to eliminate decimals.

Simplify both sides: Use the distributive property to remove parentheses and combine like terms (terms with the same variable and exponent).

Isolate the variable term: Use the addition or subtraction property of equality to move all terms containing the variable to one side of the equation, and all constant terms to the other side.

Isolate the variable: If the variable is multiplied by a coefficient, divide both sides of the equation by that coefficient (using the multiplication or division property of equality).

Simplify and check: Simplify the answer and substitute the value back into the original equation to verify that it is a solution.

Example: Solve 3x + 6 = 18.
No fractions or decimals.
No parentheses to remove or like terms to combine.
Isolate the variable term: Subtract 6 from both sides.
3x + 6 - 6 = 18 - 6
3x = 12
Isolate the variable: Divide both sides by 3.
3x/3 = 12/3
x = 4
Check: Substitute x = 4 back into the original equation.
3(4) + 6 = 18
12 + 6 = 18
18 = 18 (True)
Therefore, x = 4 is the solution.

Real-life applications

Linear equations in one variable are used in various real-life scenarios to find a single, specific unknown value, such as:

Age problems: Determining a person's age based on relationships with others' ages.

Calculating costs: Figuring out the price of an item given a total cost and fixed charges.
Distance, speed, and time problems: Calculating one of these quantities when the others are known and the speed is constant.

Solving number puzzles: Finding an unknown number based on given conditions.

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