Algebra calculator
The Algebra Calculator helps you solve algebraic equations and expressions step-by-step. Whether you’re solving linear equations, quadratic equations, systems of equations, or simplifying expressions, this calculator provides detailed solutions to help you understand the process. Select the type of problem below, enter your equation, and get instant solutions with step-by-step explanations.
Algebra Calculator
Solve linear equations of the form ax + b = c. Enter the coefficients and constants.
| Coefficient a | Coefficient b | Constant c |
Equation: 2x + 3 = 11
Solve quadratic equations of the form ax² + bx + c = 0 using the quadratic formula.
| Coefficient a | Coefficient b | Constant c |
Equation: x² – 5x + 6 = 0
Solve a system of two linear equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂.
| Equation 1: a₁x + b₁y = c₁ | ||
| a₁ | b₁ | c₁ |
| Equation 2: a₂x + b₂y = c₂ | ||
| a₂ | b₂ | c₂ |
Simplify algebraic expressions and perform operations.
| Expression (e.g., 2x + 3x – 5 + 7) |
What Is Algebra?
Algebra is the branch of mathematics that uses letters (variables) and symbols to represent unknown numbers and relationships. Instead of working with specific numbers, algebra allows you to write general rules and solve problems that apply to many situations. For example, instead of asking “What is 2 + 3?” algebra asks “What is 2x + 3 = 11?” and solves for the unknown value x. This makes algebra a powerful tool for solving real-world problems in science, engineering, economics, and everyday life.
Linear Equations
A linear equation is an equation in which the highest power of the variable is 1. The standard form is ax + b = c, where a, b, and c are known numbers and x is the unknown variable. To solve a linear equation, you isolate the variable on one side of the equals sign using inverse operations (addition/subtraction and multiplication/division).
Quadratic Equations
A quadratic equation contains a variable raised to the second power (x²) and is written in the form ax² + bx + c = 0, where a ≠ 0. Quadratic equations can have 0, 1, or 2 real solutions depending on the discriminant. The most reliable method for solving quadratic equations is the quadratic formula.
Systems of Linear Equations
A system of linear equations is a set of two or more equations with the same variables. To solve a system, you find values for the variables that satisfy all equations simultaneously. Common methods include substitution, elimination, and Cramer’s rule.
Method 1: Substitution
Solve one equation for one variable, then substitute that expression into the other equation.
Method 2: Elimination
Multiply equations by constants so that one variable has opposite coefficients, then add the equations to eliminate that variable.
Method 3: Cramer’s Rule
For a 2×2 system, use determinants to find the solution directly.
Order of Operations (PEMDAS)
When simplifying algebraic expressions, follow the order of operations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
= 2(7) + 25 (parentheses first)
= 14 + 25 (exponents done, multiplication next)
= 39 (addition last)
Combining Like Terms
Like terms are terms that have the same variable raised to the same power. When simplifying expressions, combine like terms by adding or subtracting their coefficients.
= 3x + 2x + 5 – 3 (group like terms)
= 5x + 2 (combine coefficients)
Worked Examples
3x – 7 = 5
3x = 5 + 7
3x = 12
x = 4
Example 2 — Solving Quadratic Equation
x² + 2x – 3 = 0
Using the quadratic formula with a=1, b=2, c=-3:
Δ = 2² – 4(1)(-3) = 4 + 12 = 16
x = (-2 ± 4) / 2
x₁ = 1, x₂ = -3
Example 3 — Solving System
x + y = 5
2x – y = 4
Add the equations: 3x = 9, so x = 3
Substitute: 3 + y = 5, so y = 2
Solution: (3, 2)
Properties of Algebra
| Property | Definition | Example |
|---|---|---|
| Commutative (Addition) | a + b = b + a | 3 + 5 = 5 + 3 |
| Commutative (Multiplication) | a × b = b × a | 4 × 2 = 2 × 4 |
| Associative (Addition) | (a + b) + c = a + (b + c) | (1 + 2) + 3 = 1 + (2 + 3) |
| Distributive | a(b + c) = ab + ac | 2(3 + 4) = 6 + 8 |
| Inverse (Addition) | a + (-a) = 0 | 5 + (-5) = 0 |
| Inverse (Multiplication) | a × (1/a) = 1 | 4 × 0.25 = 1 |
Frequently Asked Questions
What is the difference between an expression and an equation?
An expression is a mathematical phrase without an equals sign (e.g., 2x + 3). An equation is a statement that two expressions are equal (e.g., 2x + 3 = 11). Equations can be solved for unknown variables; expressions can only be simplified.
How do I know if a quadratic has two, one, or no real solutions?
Use the discriminant (Δ = b² – 4ac). If Δ > 0, there are two real solutions. If Δ = 0, there is one real solution. If Δ < 0, there are no real solutions (only complex ones).
Can I solve any equation algebraically?
Most equations taught in algebra (linear, quadratic, some cubic) can be solved algebraically. However, some equations require numerical methods or approximation. Higher-degree polynomials may not have closed-form solutions.
What does it mean to “isolate the variable”?
Isolating the variable means using inverse operations to get the variable by itself on one side of the equation. For example, in 2x + 3 = 11, you subtract 3 and divide by 2 to isolate x.
Why do we need to know algebra if calculators exist?
Algebra teaches problem-solving, logical thinking, and how to represent real-world situations mathematically. Understanding the process is more valuable than just getting an answer, and algebra is fundamental to advanced math, science, and engineering.
Disclaimer: This algebra calculator is provided for educational purposes to help you understand algebraic solving techniques. Always verify your solutions and understand the underlying concepts. For complex problems, consult a mathematics textbook or tutor.