Solving Linear Equations Calculator

Linear Equation Solver (ax + b = c)

Use this calculator to find the value of 'x' in a simple linear equation of the form ax + b = c. Enter the coefficients and constants, and the calculator will provide the solution for 'x'.

Understanding Linear Equations (ax + b = c)

A linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line. The most basic form of a linear equation with one variable is often expressed as ax + b = c, where:

• x is the variable you want to solve for.
• a is the coefficient of x (a number multiplied by x).
• b is a constant term.
• c is another constant term on the other side of the equation.

How to Solve ax + b = c Algebraically

The goal is to isolate the variable x on one side of the equation. Here are the steps:

1. Subtract 'b' from both sides: This moves the constant 'b' to the right side of the equation.
   ax + b - b = c - b
ax = c - b
2. Divide both sides by 'a': This isolates 'x'.
   ax / a = (c - b) / a
x = (c - b) / a

This formula allows you to find the value of x given a, b, and c.

Important Edge Cases

There are special situations to consider when solving linear equations:

• If a = 0:
  • If 0x + b = c simplifies to b = c (e.g., 0x + 5 = 5), then any value of x will satisfy the equation. In this case, there are infinitely many solutions.
  • If 0x + b = c simplifies to b ≠ c (e.g., 0x + 5 = 7), then there is no value of x that can satisfy the equation. In this case, there are no solutions.

Examples of Solving Linear Equations

Let's look at a few examples:

Example 1: Standard Case

Solve for x in the equation: 2x + 5 = 11

• Here, a = 2, b = 5, c = 11.
• Subtract 5 from both sides: 2x = 11 - 5 → 2x = 6
• Divide by 2: x = 6 / 2 → x = 3
• Solution: x = 3

Example 2: Negative Numbers

Solve for x in the equation: -3x + 7 = -8

• Here, a = -3, b = 7, c = -8.
• Subtract 7 from both sides: -3x = -8 - 7 → -3x = -15
• Divide by -3: x = -15 / -3 → x = 5
• Solution: x = 5

Example 3: Infinite Solutions

Solve for x in the equation: 0x + 4 = 4

• Here, a = 0, b = 4, c = 4.
• Since a = 0 and b = c, there are infinitely many solutions.

Example 4: No Solutions

Solve for x in the equation: 0x + 6 = 9

• Here, a = 0, b = 6, c = 9.
• Since a = 0 and b ≠ c, there are no solutions.

Infinitely many solutions.

No solutions.

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