Algebra Calculator: Solve Equations And Simplify Expressions With Steps

Algebra is the part of math that uses letters to stand for unknown numbers. This guide explains how to use the algebra calculator above and walks through the core skills it handles: simplifying, solving equations and inequalities, factoring and solving systems. Every example was checked in Python.

How To Use The Algebra Calculator

The algebra calculator above has three tabs.

1. Solve: type an equation like 3x + 7 = 22, an inequality like −2x + 5 > 11, or two equations for a system. You get the answer and each step.
2. Simplify, Expand and Factor: type an expression without an equals sign, such as 3x + 5 − x + 2(x − 4), and pick what you want to do with it.
3. Check My Answer: type an equation and your answer. The tool plugs it in and shows whether both sides match.

Type multiplication as 3x or 3*x, exponents as x^2, and fractions with parentheses, such as (x + 1)/3. Parentheses make sure the calculator reads the problem the way you mean it.

Algebra Basics: Variables, Expressions And Equations

A few words come up in every algebra problem. Knowing them makes the steps easier to follow.

You simplify an expression and solve an equation. An expression has no single answer, but an equation usually does.

How To Simplify Algebraic Expressions

To simplify an expression, remove parentheses with the distributive property, then combine like terms.

Example: simplify 3x + 5 − x + 2(x − 4)

1. Distribute the 2: 2(x − 4) = 2x − 8.
2. Rewrite: 3x + 5 − x + 2x − 8.
3. Combine the x terms: 3x − x + 2x = 4x.
4. Combine the numbers: 5 − 8 = −3.
5. Answer: 4x − 3.

Only like terms combine. You can add 3x and 2x, but not 3x and 2x², because the powers differ. And 3x + 5 stays as it is, since x and 5 are not like terms.

How To Solve Linear Equations

To solve a linear equation, get the variable alone on one side by doing the same thing to both sides. Undo addition and subtraction first, then multiplication and division.

One-step and two-step: 3x + 7 = 22

1. Subtract 7 from both sides: 3x = 15.
2. Divide both sides by 3: x = 5.
3. Check: 3(5) + 7 = 22. ✔

Variables on both sides: 5(x − 2) = 3x + 4

1. Distribute: 5x − 10 = 3x + 4.
2. Subtract 3x from both sides: 2x − 10 = 4.
3. Add 10: 2x = 14.
4. Divide by 2: x = 7.

With fractions: (1/2)x + 3 = (2/3)x − 1

1. Multiply every term by 6, the least common denominator, to clear the fractions: 3x + 18 = 4x − 6.
2. Subtract 3x: 18 = x − 6.
3. Add 6: x = 24.
4. Check: (1/2)(24) + 3 = 15, and (2/3)(24) − 1 = 15. ✔

Clearing fractions first avoids most arithmetic slips. If you need help finding a common denominator, the fraction calculator can do it.

No solution or every solution. Sometimes the variable cancels out. If x + 1 = x + 2, subtracting x leaves 1 = 2, which is false, so there is no solution. If 2(x + 1) = 2x + 2, both sides are the same, so every value of x works.

How To Solve Inequalities

Inequalities are solved like equations, with one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Example: −2x + 5 > 11

1. Subtract 5: −2x > 6.
2. Divide by −2 and flip the sign: x < −3.

The answer is a range, not one number. In interval notation, it is (−∞, −3). On a number line, it is an open circle at −3 with an arrow to the left. An open circle means −3 is not included. A closed circle is used for ≤ or ≥.

Check with a test value: try x = −4, which is less than −3. Then −2(−4) + 5 = 13, and 13 > 11. ✔

How To Expand And Factor Expressions

Expanding and factoring are opposites. Expanding multiplies out parentheses, and factoring writes an expression as a product.

Expanding with FOIL: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15. FOIL stands for first, outer, inner, last.

Squaring a binomial: (2x − 1)² = (2x − 1)(2x − 1) = 4x² − 4x + 1. A common mistake is writing 4x² + 1, which drops the middle term.

Factoring patterns:

How to factor x² − 5x + 6: find two numbers that multiply to 6 and add to −5. Those are −2 and −3, so the answer is (x − 2)(x − 3).

Always pull out the greatest common factor first.

How To Solve Quadratic Equations

A quadratic equation has an x² term. The most common ways to solve one are factoring and the quadratic formula.

By factoring: x² − 5x + 6 = 0

1. Factor: (x − 2)(x − 3) = 0.
2. Set each factor to 0: x − 2 = 0 or x − 3 = 0.
3. Solve: x = 2 or x = 3.

This works because of the zero product rule: if two numbers multiply to 0, at least one of them is 0.

When a quadratic does not factor neatly, use the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. The quadratic equation solver walks through the formula with any coefficients.

How To Solve A System Of Two Equations

A system is two or more equations that share the same variables. The solution is the pair of values that makes both true. Two common methods are substitution and elimination.

Substitution: 2x + y = 7 and x − y = 2

1. Solve the second equation for x: x = y + 2.
2. Substitute into the first: 2(y + 2) + y = 7, so 3y + 4 = 7.
3. Solve: y = 1. Then x = 1 + 2 = 3.
4. Answer: (3, 1).

Elimination: 3x + 2y = 12 and x + 4y = 14

1. Multiply the first equation by 2: 6x + 4y = 24.
2. Subtract the second equation: (6x + 4y) − (x + 4y) = 24 − 14, so 5x = 10.
3. Solve: x = 2. Then 2 + 4y = 14, so y = 3.
4. Answer: (2, 3).

Use substitution when one variable is already alone or easy to isolate. Use elimination when the coefficients line up or can be made to cancel.

Common Algebra Mistakes

Most algebra errors are sign mistakes or steps done to only one side.

Summary

• Simplify by distributing, then combining like terms.
• Solve equations by doing the same step to both sides until the variable is alone.
• Flip the inequality sign when multiplying or dividing by a negative.
• Expanding and factoring undo each other. Always factor out the GCF first.
• Solve systems with substitution or elimination, and always check your answer.

Next step: type your problem into the algebra calculator at the top of this page, or browse more math tools on the Calc Solver homepage.

Frequently Asked Questions