Algebra Calculator

What Is Algebra?

Algebra is the branch of mathematics that uses letters and symbols to represent unknown values and relationships. Instead of asking "what number plus 5 equals 12?" algebra writes x + 5 = 12 and solves for x = 7. This symbolic approach allows you to express general rules, solve complex problems, and model real-world situations with equations. The calculator above solves algebraic equations, simplifies expressions, and factors polynomials step by step.

How to Solve Linear Equations?

A linear equation has variables raised only to the first power. To solve, isolate the variable by performing the same operation on both sides. For 3x + 7 = 22: subtract 7 from both sides (3x = 15), then divide by 3 (x = 5). For equations with variables on both sides like 5x - 3 = 2x + 9: subtract 2x from both sides (3x - 3 = 9), add 3 (3x = 12), divide by 3 (x = 4). The golden rule is whatever you do to one side, you must do to the other to maintain equality.

How to Simplify Algebraic Expressions?

Simplifying combines like terms and applies the distributive property. Like terms have the same variable and exponent: 3x + 5x = 8x, but 3x + 5x² cannot be combined. The distributive property expands: 2(3x + 4) = 6x + 8. Combining both: 2(3x + 4) + 5x = 6x + 8 + 5x = 11x + 8. For fractions with variables, find common denominators: x/3 + x/4 = 4x/12 + 3x/12 = 7x/12. Simplification is not solving. It reduces complexity without finding a specific value for the variable.

How to Factor Algebraic Expressions?

Factoring reverses the distributive property, rewriting an expression as a product. Common factor: 6x² + 9x = 3x(2x + 3). Difference of squares: x² - 16 = (x + 4)(x - 4). Trinomial: x² + 5x + 6 = (x + 2)(x + 3). To factor a trinomial x² + bx + c, find two numbers that multiply to c and add to b. For x² + 5x + 6: numbers that multiply to 6 and add to 5 are 2 and 3. Factoring is essential for solving equations, simplifying fractions, and finding roots of polynomials.

Systems of Equations

A system of equations contains two or more equations with the same variables. To solve, find values that satisfy all equations simultaneously. Substitution: solve one equation for a variable and substitute into the other. From y = 2x + 1 and x + y = 7: substitute to get x + 2x + 1 = 7, so 3x = 6, x = 2, y = 5. Elimination: add or subtract equations to eliminate a variable. 2x + y = 10 and x - y = 2: add them to get 3x = 12, x = 4, y = 2. Systems model real situations with multiple constraints, such as mixture problems, cost-revenue analysis, and supply-demand equilibrium.

Inequalities

Inequalities use less than, greater than, less than or equal to, and greater than or equal to instead of equals. Solving follows the same steps as equations with one critical difference: multiplying or dividing by a negative number reverses the inequality sign. For -2x greater than 6: divide by -2 and flip the sign, giving x less than -3. Solutions are ranges of values rather than single numbers, often graphed on a number line. Compound inequalities like 2 less than x less than 7 define a range where the variable must fall.

Why Does Algebra Matter?

Algebra is the gateway to all higher mathematics and quantitative fields. Finance uses algebra for interest calculations, break-even analysis, and portfolio optimization. Science uses it to express physical laws (F = ma, E = mc²). Programming uses algebraic logic in every algorithm and formula. Even everyday decisions involve algebraic thinking: comparing phone plans, calculating tips, splitting bills, and budgeting all require setting up and solving equations. Understanding algebra transforms you from someone who can only follow specific instructions into someone who can analyze novel situations and find solutions independently. Algebra is the mathematical language that turns word problems into solvable equations and gives you the tools to find answers systematically rather than by trial and error.

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