Math Papa Algebra Calculator

Linear Equation Solver (ax + b = c)

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Understanding and Solving Linear Equations

Algebra is a fundamental branch of mathematics that uses letters (variables) to represent numbers and quantities in equations and formulas. One of the most basic and important types of equations you'll encounter in algebra is the linear equation. A linear equation is an algebraic equation in which each term has an exponent of 1, and when graphed, it forms a straight line.

What is a Linear Equation?

A common form of a linear equation with one variable is 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.

The goal when solving a linear equation is to isolate the variable 'x' on one side of the equation, determining its value.

How to Solve ax + b = c

Solving a linear equation involves a series of steps to manipulate the equation while maintaining its balance. The key principle is that whatever operation you perform on one side of the equation, you must also perform on the other side.

1. Isolate the term with 'x': To get the ax term by itself, you need to eliminate the constant b from the left side. You do this by subtracting b from both sides of the equation:
   ax + b - b = c - b
ax = c - b
2. Solve for 'x': Now that ax is isolated, you need to get 'x' by itself. Since 'a' is multiplied by 'x', you perform the inverse operation: divide both sides by a:
   ax / a = (c - b) / a
x = (c - b) / a

Important Considerations (Edge Cases)

• If 'a' is zero (a = 0):
  • If 0x + b = c simplifies to b = c (e.g., 0x + 5 = 5), then the equation is true for any value of x. This means there are infinite solutions.
  • If 0x + b = c simplifies to b ≠ c (e.g., 0x + 5 = 10), then the equation is false for any value of x. This means there is no solution.

Example Calculation

Let's use the calculator above with an example:

• Coefficient 'a': 2
• Constant 'b': 5
• Constant 'c': 15

The equation is: 2x + 5 = 15

1. Subtract 5 from both sides:
   2x + 5 - 5 = 15 - 5
2x = 10
2. Divide both sides by 2:
   2x / 2 = 10 / 2
x = 5

Using the calculator, you would input 2 for 'a', 5 for 'b', and 15 for 'c', and it would output x = 5.0000.

This linear equation solver is a handy tool for quickly finding the value of 'x' in simple algebraic expressions, helping you verify your manual calculations or explore different scenarios.