2.1-Introduction to Linear Equations in One Variable

2.1-Introduction to Linear Equations in One Variable Important Formulae

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2.1 Introduction to Linear Equations in One Variable
  • A linear equation in one variable is an equation of the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable.
  • The degree of the equation is 1, meaning the highest power of the variable is 1.
  • To solve the equation, the goal is to isolate the variable $x$ on one side of the equation.
  • Example: Solve $3x - 5 = 10$.
  • To solve, first add 5 to both sides: $3x = 15$.
  • Then divide both sides by 3: $x = 5$.

2.1-Introduction to Linear Equations in One Variable

A linear equation in one variable is an equation of the form:

ax + b = 0, where a and b are constants, and x is the variable. The goal is to solve for the value of x.

In a linear equation, the degree of the variable is always 1, meaning the highest power of x is 1. The equation represents a straight line when plotted on a graph, which is why it is referred to as a "linear" equation.

The equation ax + b = 0 can also be written as:

x = -b/a, where a ≠ 0. This is the general form of the solution for a linear equation in one variable.

Examples of Linear Equations in One Variable:
  • 2x - 4 = 0
  • 5x + 3 = 10
  • -3x = 9
  • x/4 + 2 = 5

To solve these equations, we use basic algebraic operations to isolate the variable x.

Steps to Solve Linear Equations in One Variable:
  1. Identify the equation and ensure it is in the standard form ax + b = 0.
  2. Move the constant term to the other side by performing the opposite operation (adding or subtracting).
  3. Isolate the variable x by dividing or multiplying by the coefficient of x (which is a in the equation ax + b = 0).
  4. Simplify to find the value of x.
Example 1:

Solve 2x - 4 = 0.

Step 1: Add 4 to both sides: 2x = 4.

Step 2: Divide both sides by 2: x = 2.

Thus, the solution is x = 2.

Example 2:

Solve 5x + 3 = 10.

Step 1: Subtract 3 from both sides: 5x = 7.

Step 2: Divide both sides by 5: x = 7/5 or x = 1.4.

Example 3:

Solve -3x = 9.

Step 1: Divide both sides by -3: x = -3.

Thus, the solution is x = -3.

Properties of Linear Equations in One Variable:
  • Linear equations in one variable have only one solution, i.e., a unique value for x.
  • The solution can be an integer, fraction, or decimal, depending on the equation.
  • Linear equations can also be expressed in fractional or decimal form.

Linear equations form the foundation for more complex algebraic concepts and help develop problem-solving skills. Mastery of solving linear equations in one variable is crucial for progressing in algebra and other branches of mathematics.