Linear Equation Solver

Linear Solver & Step-by-Step Isolation Analyzer

What is a Linear Equation?

In algebra, a linear equation is defined as an equation where the highest exponent of any variable (such as , , or any letter from a to z) is 1.

Graphing a linear equation always results in a straight line. These equations are typically categorized into two types:

  • Single Variable: (e.g., ). Used to find a specific "Root" or numerical value for the unknown.
  • Dual Variable: (e.g., ). Used to describe an infinite set of coordinate points that form a line path.

The Geometric Connection

Algebra and Geometry are linked. When we solve for a single variable, we find a point. In a 2D plane, equations like $x=5$ appear as vertical lines, while $y=5$ appears as horizontal lines. When we solve an $x$ and $y$ equation, we describe every possible solution as a diagonal line.

The X-Intercept is the "Root" of the equation—where the line crosses the horizontal axis ($y = 0$). The Y-Intercept is where the line crosses the vertical axis ($x = 0$).

The Logic of Solving Linear Equations

Solving linear equations is built on the Golden Rule of Algebra: Whatever operation you perform on one side of the equation, you must perform on the other side to keep it balanced.

  1. Combine Like Terms: Simplify each side of the equal sign individually first.
  2. Group Variables: Move all terms containing the target variable to one side. Remember, if you subtract it from the right, you must subtract it from the left.
  3. Isolate Constants: Move all numbers without variables to the opposite side using the same balancing principle.
  4. Final Division: Divide both sides by the variable's coefficient to find the final value or functional form.

The "Check Step": Verifying Your Work

One of the most important habits in algebra is verifying your solution. Once you find a value for your variable, you should plug it back into the original equation to see if it makes the statement true.

  • If both sides are equal (e.g., ), your answer is correct!
  • If the sides are unequal (e.g., ), a mistake was made during isolation.

Mastering Linear Equations: Worked Examples

Explore these detailed examples to see exactly how equations are balanced using the Golden Rule.

Example 1: Basic Two-Step Equation

Problem: Solve for n in

Step 1: Subtract 5 from both sides.

Result:

Step 2: Divide both sides by 3.

Final Answer:

Example 2: Slope-Intercept Form (x and y)

Problem: Rearrange into functional form.

Step: Subtract 2x from both sides to group the variables.

Final Answer:

Frequently Asked Questions

Why use vertical fractions instead of decimals?

Vertical fractions are the standard textbook representation of exact values. Decimals are often rounded approximations (like $0.333$ for $1/3$), whereas fractions maintain total precision.

What is the "Root" of a line?

The "Root" is just another name for the x-intercept. It is the value of $x$ that makes the function equal to zero ($y = 0$).