Algebraic expressions are combinations of numbers, variables (like x, y, z), and mathematical operations (addition, subtraction, multiplication, division). Simplifying these expressions is a fundamental skill in algebra, making them easier to understand, analyze, and use in further calculations. This process involves combining like terms and applying the order of operations.
Variables: Symbols representing unknown values (e.g., x, a, height).
Terms: Parts of an expression separated by addition or subtraction. A term can be a number, a variable, or a product of numbers and variables (e.g., in 3x + 5, 3x and 5 are terms).
Coefficients: The numerical factor multiplying a variable in a term (e.g., in 3x, 3 is the coefficient).
Why Simplify?
Simplifying an algebraic expression means rewriting it in its most compact and understandable form without changing its value. This is crucial for:
Making complex equations manageable.
Solving equations more efficiently.
Analyzing patterns and relationships.
Preparing expressions for graphing or further mathematical manipulation.
How to Simplify: Combining Like Terms
The most common method for simplifying expressions is by combining like terms. Like terms are terms that have the exact same variable part raised to the exact same power. For example, 3x and -2x are like terms, but 3x and 3x² are not.
To combine like terms, you simply add or subtract their coefficients. The variable part remains the same.
Identify like terms: The terms with x are 3x and -2x. The terms with y are 5y. The constant term is 7.
Combine the 'x' terms:3x - 2x = (3 - 2)x = 1x, which is usually written as x.
Combine the 'y' terms: There's only one term with y, which is 5y.
Combine the constant terms: There's only one constant term, 7.
Write the simplified expression: Combine the results: x + 5y + 7.
More Complex Scenarios
Simplification can also involve:
Distributive Property: When parentheses are present, like in 2(x + 3) + 5x. You first distribute the 2 to get 2x + 6, then combine like terms: 2x + 6 + 5x = 7x + 6.
Fractions and Exponents: These require adherence to the order of operations (PEMDAS/BODMAS) and specific rules for manipulating algebraic fractions and exponents.
This calculator helps automate the process of combining like terms for basic algebraic expressions. For more complex scenarios involving distribution, exponents, or fractions, manual application of algebraic rules is necessary.
function simplifyExpression() {
var expressionInput = document.getElementById("expression");
var expression = expressionInput.value.trim();
var resultDiv = document.getElementById("simplified-expression");
var resultArea = document.getElementById("result-area");
if (expression === "") {
resultDiv.textContent = "Please enter an expression.";
resultArea.style.display = "block";
return;
}
// Basic parser and simplifier for linear expressions with x, y, z and constants
// This is a simplified parser and won't handle complex math like exponents, fractions, etc.
// It focuses on combining like terms for variables x, y, z and constants.
var terms = {}; // Object to store coefficients for each variable type
terms['constant'] = 0; // Initialize constant term
// Regular expression to find terms: optional sign, optional coefficient, variable, optional exponent (only for basic x, y, z)
// It looks for patterns like: +3x, -5y, 2z, 7, -x, +y
var termRegex = /([+-]?\s*\d*\.?\d*)([a-zA-Z])?/g;
var matches;
// Add implicit '+' at the beginning if the expression doesn't start with '+' or '-'
if (!expression.startsWith('+') && !expression.startsWith('-')) {
expression = '+' + expression;
}
// Add '+' before a term if it starts with a variable without a sign (e.g., x+5 -> +x+5)
// Ensure it doesn't add double signs like ++x
expression = expression.replace(/([a-zA-Z])/g, function(match, p1, offset, string) {
if (offset > 0 && (string[offset – 1] !== '+' && string[offset – 1] !== '-')) {
return '+' + p1;
}
return p1;
});
while ((matches = termRegex.exec(expression)) !== null) {
if (matches[0] === "" || matches[0].trim() === "") continue; // Skip empty matches
var fullTerm = matches[0].trim();
var coefficientStr = matches[1];
var variable = matches[2];
var coefficient = 0;
if (coefficientStr === '+' || coefficientStr === ") { // Handles cases like '+x' or 'x'
coefficient = 1;
} else if (coefficientStr === '-') { // Handles case like '-x'
coefficient = -1;
} else {
// Remove leading '+' if present for parsing
if (coefficientStr.startsWith('+')) {
coefficientStr = coefficientStr.substring(1);
}
coefficient = parseFloat(coefficientStr);
if (isNaN(coefficient)) { // Handle cases like just '+', '-', or invalid numbers
if (variable) { // If it's a variable like '+' or '-'
coefficient = (coefficientStr === '+') ? 1 : -1;
} else { // If it's just a number like '+' or '-'
coefficient = parseFloat(coefficientStr);
}
}
}
if (isNaN(coefficient)) {
// This might happen for malformed inputs, try to recover
// If we have a variable but no number, assume 1/-1
if (variable && (coefficientStr === '+' || coefficientStr === '-')) {
coefficient = (coefficientStr === '+') ? 1 : -1;
} else {
console.warn("Could not parse coefficient for term:", fullTerm);
continue; // Skip this term if parsing fails
}
}
if (variable) {
// Initialize variable in terms if not present
if (!terms[variable]) {
terms[variable] = 0;
}
terms[variable] += coefficient;
} else {
// It's a constant term
terms['constant'] += coefficient;
}
}
var simplifiedParts = [];
var variables = Object.keys(terms).sort(); // Sort variables alphabetically for consistent output
for (var i = 0; i 0 && coeff > 0) {
simplifiedParts.push("+" + part);
} else if (coeff < 0) {
// Ensure the '-' sign is handled correctly by removing it from the part if it starts with it
if(part.startsWith('-')) {
simplifiedParts.push(part);
} else {
simplifiedParts.push("-" + part.substring(1)); // Should not happen with coeff 0 && constantCoeff > 0) {
simplifiedParts.push("+" + constantCoeff);
} else if (constantCoeff 0 and simplifiedParts is empty
simplifiedParts.push(constantCoeff);
}
}
var finalExpression = simplifiedParts.join(" ");
// Post-processing for cleaner output
finalExpression = finalExpression.replace(/\+ -/g, '-'); // Replace '+ -' with '-'
finalExpression = finalExpression.replace(/^- /, '-'); // Remove leading '+ ' if expression starts positive
finalExpression = finalExpression.replace(/^ \+/, "); // Remove leading '+ ' if expression starts positive
finalExpression = finalExpression.replace(/^([\w\d]+)$/, '$1'); // Remove space if only one term
finalExpression = finalExpression.replace(/([\w\d]+) \+/, '$1+'); // Remove space before '+' if term is first
finalExpression = finalExpression.replace(/([\w\d]+)$/, '$1'); // Ensure no trailing spaces
if (finalExpression === "" || finalExpression.trim() === "") {
finalExpression = "0"; // If all terms cancelled out
}
// Clean up potential leading '+' after processing
if (finalExpression.startsWith('+')) {
finalExpression = finalExpression.substring(1).trim();
}
resultDiv.textContent = finalExpression;
resultArea.style.display = "block";
}