Linear Equations in One Variable

What is the condition for a linear equation in one variable to have a unique solution?

What is the first step in solving a linear equation?

What is the graphical representation of a linear equation in one variable?

What is the slope of the line representing a linear equation in one variable?

What happens to a linear equation in one variable if a = 0 and b = c?

What is the purpose of dividing both sides of the equation by the coefficient of the variable?

Definition and Notation

A linear equation in one variable is an equation that can be written in the form ax + b = c, where:
- a, b, and c are constants
- a is the coefficient of the variable x
- x is the variable

Properties

A linear equation in one variable has a unique solution if a ≠ 0
If a = 0, the equation is a linear equation in one variable only if b = c, in which case it has infinitely many solutions
If a = 0 and b ≠ c, the equation has no solution

Solving Linear Equations

To solve a linear equation, isolate the variable x by performing the following steps:
1. Add or subtract the same value to both sides of the equation to isolate the term with the variable
2. Divide both sides of the equation by the coefficient of the variable (a) to solve for x
Example: 2x + 3 = 7
- Subtract 3 from both sides: 2x = 4
- Divide both sides by 2: x = 2

Graphical Representation

The graph of a linear equation in one variable is a straight line on the number line
The line intersects the x-axis at the solution to the equation
The slope of the line is the coefficient of the variable (a)

Learn about the definition, properties, and solving methods of linear equations in one variable. Understand how to isolate the variable and graph the equation on the number line.