Simplify Equations Calculator

Linear Equation Simplifier

Use this calculator to simplify and solve linear equations of the form Ax + B = Cx + D for the variable x. Enter the coefficients and constants for your equation below.

function calculateEquationSimplification() { var coeffA = parseFloat(document.getElementById('coeffA').value); var constB = parseFloat(document.getElementById('constB').value); var coeffC = parseFloat(document.getElementById('coeffC').value); var constD = parseFloat(document.getElementById('constD').value); var resultDiv = document.getElementById('equationResult'); if (isNaN(coeffA) || isNaN(constB) || isNaN(coeffC) || isNaN(constD)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } // Equation: Ax + B = Cx + D // Rearrange to: Ax – Cx = D – B // Factor x: (A – C)x = D – B // Solve for x: x = (D – B) / (A – C) var leftSideCoeffDiff = coeffA – coeffC; // (A – C) var rightSideConstDiff = constD – constB; // (D – B) if (leftSideCoeffDiff === 0) { // Case 1: Coefficients of x are equal (A = C) if (rightSideConstDiff === 0) { // Case 1a: Constants are also equal (B = D), so 0 = 0 resultDiv.innerHTML = "Infinite Solutions: The equation simplifies to 0 = 0, meaning any value of x satisfies the equation."; } else { // Case 1b: Constants are not equal (B != D), so 0 = (non-zero number) resultDiv.innerHTML = "No Solution: The equation simplifies to 0 = " + rightSideConstDiff + ", which is false. There is no value of x that satisfies the equation."; } } else { // Case 2: Coefficients of x are not equal (A != C) var xValue = rightSideConstDiff / leftSideCoeffDiff; resultDiv.innerHTML = "x = " + xValue.toFixed(4) + ""; } } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 600px; margin: 30px auto; border: 1px solid #e0e0e0; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { color: #555; margin-bottom: 25px; line-height: 1.6; text-align: center; } .form-group { margin-bottom: 18px; display: flex; flex-direction: column; } .form-group label { margin-bottom: 8px; color: #444; font-weight: bold; font-size: 0.95em; } .form-group input[type="number"] { padding: 12px; border: 1px solid #ccc; border-radius: 6px; font-size: 1em; width: 100%; box-sizing: border-box; transition: border-color 0.3s ease; } .form-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.2); } button { background-color: #007bff; color: white; padding: 13px 25px; border: none; border-radius: 6px; cursor: pointer; font-size: 1.1em; font-weight: bold; width: 100%; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 15px; } button:hover { background-color: #0056b3; transform: translateY(-2px); } button:active { transform: translateY(0); } .result-container { background-color: #e9f7ff; border: 1px solid #b3e0ff; border-radius: 8px; padding: 20px; margin-top: 30px; text-align: center; } .result-container h3 { color: #0056b3; margin-top: 0; margin-bottom: 15px; font-size: 1.4em; } #equationResult { color: #333; font-size: 1.2em; font-weight: bold; word-wrap: break-word; }

Understanding and Simplifying Linear Equations

Simplifying equations is a fundamental skill in mathematics, allowing us to reduce complex expressions into simpler, more manageable forms or to solve for unknown variables. While "simplifying" can mean many things depending on the type of equation (e.g., combining like terms, factoring, expanding), one of the most common forms of simplification involves solving for a specific variable in a linear equation.

What is a Linear Equation?

A linear equation is an algebraic equation in which each term has an exponent of 1, and the graph of the equation is a straight line. It typically involves one or more variables, but in its simplest form, it can be written as Ax + B = 0, where A and B are constants and x is the variable.

The Standard Form for Our Calculator

Our calculator focuses on simplifying and solving linear equations presented in the form:

Ax + B = Cx + D

Here:

• A and C are the coefficients of the variable x on the left and right sides of the equation, respectively.
• B and D are constant terms on the left and right sides of the equation, respectively.

The goal of simplifying this type of equation is to isolate the variable x on one side of the equation, thereby finding its numerical value.

How to Simplify and Solve Ax + B = Cx + D

The process involves a few algebraic steps:

1. Gather x-terms on one side: Subtract Cx from both sides of the equation to move all terms containing x to the left side.
   Ax - Cx + B = D
2. Gather constant terms on the other side: Subtract B from both sides of the equation to move all constant terms to the right side.
   Ax - Cx = D - B
3. Factor out x: Factor x from the terms on the left side.
   (A - C)x = D - B
4. Isolate x: Divide both sides by (A - C) to solve for x.
   x = (D - B) / (A - C)

Understanding Special Cases

While the formula x = (D - B) / (A - C) works for most linear equations, there are two important special cases to consider:

• No Solution: This occurs when the coefficients of x are equal (A = C), but the constants are different (B ≠ D). In this scenario, the equation simplifies to a false statement, such as 0 = 5. This means there is no value of x that can satisfy the equation.
  Example: If 3x + 5 = 3x + 7, then (3-3)x = (7-5) simplifies to 0x = 2, or 0 = 2, which is impossible.
• Infinite Solutions: This happens when both the coefficients of x are equal (A = C) AND the constants are also equal (B = D). The equation simplifies to a true statement, such as 0 = 0. This indicates that any real number for x will satisfy the equation.
  Example: If 4x + 6 = 4x + 6, then (4-4)x = (6-6) simplifies to 0x = 0, or 0 = 0, which is always true.

Examples Using the Calculator

Let's walk through a few examples to demonstrate how to use the calculator and interpret its results:

Example 1: Standard Solution

Equation: 3x + 5 = 2x + 10

• Coefficient A: 3
• Constant B: 5
• Coefficient C: 2
• Constant D: 10

Calculation:
x = (10 - 5) / (3 - 2)
x = 5 / 1
x = 5

Calculator Result: x = 5.0000

Example 2: No Solution

Equation: 2x + 4 = 2x + 5

• Coefficient A: 2
• Constant B: 4
• Coefficient C: 2
• Constant D: 5

Calculation:
(2 - 2)x = (5 - 4)
0x = 1
0 = 1 (False)

Calculator Result: No Solution: The equation simplifies to 0 = 1, which is false. There is no value of x that satisfies the equation.

Example 3: Infinite Solutions

Equation: 4x + 7 = 4x + 7

• Coefficient A: 4
• Constant B: 7
• Coefficient C: 4
• Constant D: 7

Calculation:
(4 - 4)x = (7 - 7)
0x = 0
0 = 0 (True)

Calculator Result: Infinite Solutions: The equation simplifies to 0 = 0, meaning any value of x satisfies the equation.

By using this calculator, you can quickly determine the solution to various linear equations, helping you to understand the principles of algebraic simplification and problem-solving.