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Linear Equation Simplifier: Solve for X in ax + b = c

Understanding and solving linear equations is a fundamental skill in mathematics, crucial for various fields from science and engineering to finance. 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 common form for a single-variable linear equation is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for.

What is a Linear Equation?

A linear equation is an equation that can be written in the form Ax + By = C for two variables, or ax + b = c for a single variable. The key characteristic is that the highest power of the variable(s) is 1. When plotted on a graph, a linear equation always forms a straight line. Solving a linear equation means finding the value(s) of the variable(s) that make the equation true.

Why Simplify and Solve Linear Equations?

Simplifying and solving linear equations allows us to determine unknown quantities based on known relationships. For instance, if you know the cost per item (a), a fixed fee (b), and the total amount spent (c), you can use a linear equation to find out how many items (x) were purchased. This process of isolating the variable 'x' is often referred to as simplifying the equation to find its solution.

How to Solve ax + b = c

To solve for 'x' in the equation ax + b = c, we follow these algebraic steps:

  1. Isolate the term with 'x': Subtract 'b' from both sides of the equation.
    ax + b - b = c - b
    ax = c - b
  2. Isolate 'x': Divide both sides of the equation by 'a'.
    ax / a = (c - b) / a
    x = (c - b) / a

There are special cases to consider:

  • If a = 0 and c - b = 0 (meaning 0 = 0), then there are infinitely many solutions for 'x'. Any value of 'x' will satisfy the equation.
  • If a = 0 and c - b ≠ 0 (meaning 0 = (a non-zero number)), then there is no solution for 'x'. The equation is contradictory.

Using the Linear Equation Simplifier

Our calculator helps you quickly find the value of 'x' for any linear equation in the form ax + b = c. Simply input the values for 'a', 'b', and 'c' into the fields below, and the calculator will instantly provide the solution for 'x', along with handling the special cases.

Solve ax + b = c for X

function calculateLinearEquation() { var a = parseFloat(document.getElementById('coefficientA').value); var b = parseFloat(document.getElementById('constantB').value); var c = parseFloat(document.getElementById('constantC').value); var resultDiv = document.getElementById('result'); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; return; } var output = "; if (a === 0) { // Case: 0x + b = c => b = c if (b === c) { output = 'Equation: ' + b + ' = ' + c + "; output += 'Since ' + b + ' equals ' + c + ', this equation simplifies to 0 = 0.'; output += 'Result: Infinitely many solutions for X.'; } else { output = 'Equation: ' + b + ' = ' + c + "; output += 'Since ' + b + ' does not equal ' + c + ', this equation is contradictory.'; output += 'Result: No solution for X.'; } } else { // Standard case: ax + b = c var x = (c – b) / a; output = 'Original Equation: ' + a + 'x + ' + b + ' = ' + c + "; output += 'Step 1: Subtract ' + b + ' from both sides to isolate the term with x.'; output += " + a + 'x = ' + (c – b) + "; output += 'Step 2: Divide both sides by ' + a + ' to solve for x.'; output += 'x = (' + (c – b) + ') / ' + a + "; output += 'Result: X = ' + x.toFixed(4) + "; // Display with 4 decimal places } resultDiv.innerHTML = output; } // Initial calculation on page load for default values window.onload = calculateLinearEquation;

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