Algebra Calculator with Steps for Free

Linear Equation Solver: ax + b = c

Result:

Steps:

function calculateLinearEquation() { var coeffA = parseFloat(document.getElementById('coeffA').value); var constB = parseFloat(document.getElementById('constB').value); var constC = parseFloat(document.getElementById('constC').value); var resultX = document.getElementById('resultX'); var stepsOutput = document.getElementById('stepsOutput'); resultX.innerHTML = "; stepsOutput.innerHTML = "; if (isNaN(coeffA) || isNaN(constB) || isNaN(constC)) { resultX.innerHTML = 'Please enter valid numbers for all fields.'; resultX.style.color = '#dc3545'; return; } if (coeffA === 0) { if (constB === constC) { resultX.innerHTML = 'Infinite solutions (0x = 0).'; } else { resultX.innerHTML = 'No solution (0x = ' + (constC – constB) + ').'; } resultX.style.color = '#dc3545'; stepsOutput.innerHTML = 'Starting equation: ' + coeffA + 'x + ' + constB + ' = ' + constC + '\n' + 'If a = 0, the equation becomes: ' + constB + ' = ' + constC + '\n' + 'This is either always true (infinite solutions) or always false (no solution).'; return; } var steps = []; var currentEquation = "; // Step 1: Original Equation currentEquation = coeffA + 'x + ' + constB + ' = ' + constC; steps.push('Step 1: Original Equation'); steps.push(' ' + currentEquation); // Step 2: Subtract 'b' from both sides var cMinusB = constC – constB; currentEquation = coeffA + 'x = ' + cMinusB; steps.push('\nStep 2: Subtract ' + constB + ' from both sides'); steps.push(' ' + coeffA + 'x + ' + constB + ' – ' + constB + ' = ' + constC + ' – ' + constB); steps.push(' ' + currentEquation); // Step 3: Divide by 'a' var xValue = cMinusB / coeffA; currentEquation = 'x = ' + xValue; steps.push('\nStep 3: Divide both sides by ' + coeffA); steps.push(' ' + coeffA + 'x / ' + coeffA + ' = ' + cMinusB + ' / ' + coeffA); steps.push(' ' + currentEquation); resultX.innerHTML = 'x = ' + xValue.toFixed(4); // Display with 4 decimal places resultX.style.color = '#28a745'; stepsOutput.innerHTML = steps.join('\n'); }

Understanding Linear Equations and How to Solve Them

Algebra is a fundamental branch of mathematics that uses letters (variables) to represent numbers and quantities in equations. One of the most common types of equations you'll encounter is a linear equation. A linear equation is an algebraic equation in which each term has an exponent of 1, and when plotted on a graph, it forms a straight line. The most basic form of a linear equation with one variable is ax + b = c.

What is a Linear Equation?

In the equation ax + b = c:

  • x is the variable you are trying to solve for.
  • a is the coefficient of x. It's the number that multiplies the variable.
  • b is a constant term.
  • c is another constant term on the other side of the equals sign.

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

How to Solve ax + b = c Manually

Solving a linear equation involves a series of inverse operations to maintain balance while isolating the variable. Here are the steps:

  1. Start with the original equation: ax + b = c
  2. Isolate the term with 'x': To get rid of the constant b on the left side, you perform the inverse operation. Since b is being added, you subtract b from both sides of the equation.
    ax + b - b = c - b
    This simplifies to: ax = c - b
  3. Isolate 'x': Now, x is being multiplied by a. To isolate x, you perform the inverse operation of multiplication, which is division. Divide both sides of the equation by a.
    ax / a = (c - b) / a
    This simplifies to: x = (c - b) / a

Important Note: If a is equal to zero, you cannot divide by a. In this special case:

  • If 0x + b = c simplifies to b = c (e.g., 5 = 5), then there are infinitely many solutions for x.
  • If 0x + b = c simplifies to b ≠ c (e.g., 5 = 7), then there is no solution for x.

Example Walkthrough: 2x + 5 = 11

Let's use the calculator's default values to demonstrate the manual steps:

  1. Original Equation: 2x + 5 = 11
  2. Subtract 5 from both sides:
    2x + 5 - 5 = 11 - 5
    2x = 6
  3. Divide both sides by 2:
    2x / 2 = 6 / 2
    x = 3

So, the solution to the equation 2x + 5 = 11 is x = 3.

Using the Calculator

Our free online linear equation solver simplifies this process. Simply input the coefficient 'a', constant 'b', and constant 'c' into the respective fields. Click the "Solve Equation" button, and the calculator will instantly provide you with the value of 'x' and a detailed breakdown of each step taken to reach the solution. This tool is perfect for checking your homework, understanding the process, or quickly solving equations without manual calculation.

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