Solve the Equation Algebraically Calculator

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Linear Equation Solver (ax + b = c)

Enter the coefficients and constants for your linear equation in the form ax + b = c to find the value of 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; } if (a === 0) { if (b === c) { resultDiv.innerHTML = "The equation simplifies to " + b + " = " + c + ". This statement is true, which means there are infinitely many solutions for x."; } else { resultDiv.innerHTML = "The equation simplifies to " + b + " = " + c + ". This statement is false, which means there is no solution for x."; } } else { var x = (c – b) / a; resultDiv.innerHTML = "For the equation " + a + "x + " + b + " = " + c + ", the solution is x = " + x.toFixed(4) + "."; } }

Understanding and Solving Algebraic Equations

Algebraic equations are fundamental to mathematics and are used to represent relationships between quantities. Solving an algebraic equation means finding the value(s) of the variable(s) that make the equation true. While algebra encompasses a vast array of equation types, one of the most common and foundational is the linear equation.

What is a Linear Equation?

A linear equation is an algebraic equation in which each term has an exponent of 1 and no variable is multiplied by another variable. The most basic form of a linear equation with one variable is often written as ax + b = c, where:

  • x is the variable (the unknown value we want to find).
  • a is the coefficient of x (a number that multiplies x).
  • b is a constant term (a number that stands alone).
  • c is another constant term on the other side of the equals sign.

The goal when solving such an equation is to isolate the variable x on one side of the equation.

How to Algebraically Solve ax + b = c

To solve a linear equation of the form ax + b = c, we follow a series of inverse operations to isolate x:

  1. Subtract b from both sides: This eliminates the constant term from the side with x.
    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

Examples of Solving Linear Equations

Let's walk through a few examples to illustrate the process:

Example 1: A Simple Case

Solve the equation: 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

To verify, substitute x = 5 back into the original equation: 2(5) + 5 = 10 + 5 = 15. This is true, so our solution is correct.

Example 2: Dealing with Negative Numbers

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

  1. Subtract 7 from both sides:
    -3x + 7 - 7 = -8 - 7
    -3x = -15
  2. Divide both sides by -3:
    -3x / -3 = -15 / -3
    x = 5

Verification: -3(5) + 7 = -15 + 7 = -8. Correct.

Example 3: Special Case – When a = 0

Consider the equation: 0x + 4 = 4

If a = 0, the equation becomes b = c. In this case, 4 = 4, which is a true statement. This means that any value of x will satisfy the equation. Therefore, there are infinitely many solutions.

Consider the equation: 0x + 4 = 7

Here, a = 0, and the equation becomes 4 = 7, which is a false statement. This means that no value of x can satisfy the equation. Therefore, there is no solution.

This calculator helps you quickly solve linear equations of the form ax + b = c by applying these algebraic principles, including handling the special cases where the coefficient 'a' is zero.

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