Linear Equations in One Variable Chapter 15

Study Notes

Algebraic expressions include terms like 5x, 3x + y, and x² + 1, where some contain multiple variables.
Linear equations only involve one variable and have a highest power of 1, e.g., 2x, 3y – 7.
Equations feature an equality sign (=), distinguishing them from expressions.

An algebraic equation comprises a left-hand side (LHS) and a right-hand side (RHS).
For example, in the equation 2x – 3 = 7, the LHS is 2x – 3, and RHS is 7.
Values that satisfy the equation are termed solutions; for instance, x = 5 is a solution for 2x – 3 = 7.

The balance of an equation must be maintained, applying the same mathematical operation to both sides.
Equations can have variables on both sides, e.g., 2x – 3 = x + 2, requiring manipulations to isolate the variable.

To solve the equation 2x – 3 = x + 2:
- Transpose terms to isolate x, yielding x = 5.
For more complex equations like 5x + 3/2 = x – 14:
- Multiply through by the denominator to eliminate fractions and isolate x accordingly, arriving at x = -5.

Example: Solve (6x + 1)/3 + 1 = (x - 3)/6:
- Use the least common multiple (LCM) to simplify to a standard form.
The goal is to transform complex equations by opening brackets and combining like terms, leading to simpler equations for easier solving.

Linear equations in one variable are fundamental to algebra, featuring only one variable raised to the first power.
Solutions require testing values for equality on both sides of the equation.
Simplification may involve clearing fractions, combining like terms, and transposing variables across the equation.
Linear equations are powerful tools for solving practical problems involving various scenarios, such as finance, geometry, and age-related problems.