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Algebra Calculator: Step-by-Step help to solve Algebra problems

Have you ever tried to assemble a jigsaw puzzle with missing pieces and pondered how to find which pieces fit where? Welcome into the realm of algebra! In an amazing mathematical puzzle, letters and symbols take place of unknown numbers. This fundamental branch of mathematics helps us to apply mathematical equations and formulas to visually represent real-world problems. Algebra is there to assist you in everything from determining your monthly budget to calculating how long it takes to get anywhere to even developing a computer program.

Origin of Algebra:

Algebra originated in ancient Egypt and Babylonia. Algebra comes from an Arabic term meaning "restoration" or "completion." Often credited with giving algebra its name, Diophantus in Greece, Brahmagupta in India, and al-Khwarizmi in Baghdad made significant contributions.

What is algebra?

Algebra, then, is essentially a branch of mathematics focused on variables, symbols and their operations under guidelines. Mostly letters x, y, and z, these symbols—which stand for quantities without set values—are called variables. Algebra provides general formulas and lets us solve problems for many distinct values.Fundamental ideas

Fundamental Concepts:

Variables: Symbols that represent changeable or unknown numbers.

Constants: Fixed values that do not change.

Expressions: Integration of variables, constants, and operations—like addition and multiplication.

Equations: Mathematical statements that show the equality of two expressions constitute equations.

Understanding Variables and Constants

Variables are like empty boxes that can hold any number. They're placeholders for values we don't know yet or that can change.

Example: In the expression $5x+3$, x is a variable.

Constants are numbers that have a fixed value.

Example: In the same expression 5x+3, 3 is a constant.

Variables and constants work together in expressions and equations to model real-world situations.

The Language of Algebra

Algebra has its own language and symbols:

Operations: Addition (+), subtraction (−), multiplication (× or implied by juxtaposition), and division (÷ or /).

Coefficients: Numbers multiplied by variables. In 5x, 5 is the coefficient.

Terms: The parts of an expression separated by addition or subtraction. In 3x+2, 3x and 2 are terms.

Solving algebraic problems would require understanding this language.

Simplification of Algebraic Expressions Why Simplify Expressions?

Simplifying expressions helps one to grasp and work with them. It combines similar terms and applies mathematical concepts to create as simple expressions as possible.

Combining Like Terms

Like terms are terms that have the same variables with same exponents.

Example: 7x and 3x are like terms because they both contain x.

Combine Like Terms:

Identify like terms in the expression.

Add or subtract the coefficients of like terms.

Rewrite the expression with combined terms.

Example:

Simplify 4x+5−2x+3.

Combine the like terms (4x and −2x): 4x−2x=2x. Combine constants (5 and 3; 5 + 3 = 8). Rewrite the simplified expression: 2x+8.

Distributive Property

The distributive property of multiplication helps you to remove parentheses. Distributive Property Formula: a(b+c) = ab+ac

How to Use It:

1.Multiply the term outside the parentheses by each term inside. 2.Simplify the resulting expression by combining like terms if necessary.

Example:

Simplify 3(2x + 4). a. Multiply 3 to each term inside the parentheses: 3. 2x + 3 . 4 b. Multiply: 6x + 12

Simplifying Complex Expressions

For expressions with multiple parentheses and terms, use the distributive property to multiply the terms or the constants with the terms given inside the parentheses then combine like terms step by step.

Example:

Simplify 3(x+2) + 5(x−1).

Multiply 3 to each term inside the first set of parentheses: 3. x + 3. 2 = 3x + 6

Multiply 5 to each term inside the second set of parentheses: 5. x – 5. 1 = 5x − 5

Combine the results: 3x + 6 + 5x − 5

Combine like terms: (3x + 5x) + (6 − 5) = 8x + 1

So, 3(x + 2) + 5(x − 1) simplifies to 8x + 1.

Solving Algebraic Equations What's an equation?

An equation refers to the mathematical statement that represents the balance between two expressions' equality, made with an equals sign (=). Solving an equation is determining the value(s) of the variables that satisfy it.

An equation is a mathematical statement that asserts the equality of two expressions, using an equals sign (=). Solving an equation is determining the value(s) of the variables that satisfy it.

Objective of Solving Equations The main objective is to separate the variable on one side of the equation to find its value. Solving One-Step Equations

Addition or Subtraction Equations

Example: Solve x+8=12.

Subtract 8 from both sides: x=12−8

Solution: x=4

Multiplication or Division Equations

Example: Solve 4x=24.

Divide both sides by 4: x = 24 ÷ 4.

Solution: x=6.

Solving Two-Step Equations

Example: Solve 4x-5=7. 1.Add 5 to both sides: 4x=12. 2.Divide both sides by 4: x=3

Solving Multi-Step Equations

Example: Solve 7(x−2)+3=10. 1.Distribute: 7x−14+3=10. 2.Combine like terms: 7x−11=10. 3.Add 11 to both sides: 7x=21. 4.Divide by 7: x=3.

Solving equations with variables on both sides

Example: Solve 5x+2=x+10. 1.Subtract x from both sides: 5x−x+2=10. 2.Simplify: 4x+2=10. 3.Subtract 2 from both sides: 4x = 8. 4.Divide by 4 on both sides: x = 2.

Verifying the Solution Substitute your solution into the original equation to verify that it satisfies the equation.

Check: Does 5(2)+2=2+10?

Left Side: 10+2=12

Right Side: 12=12

Both sides are equal, so x=2 is correct.

Understanding Inequalities What Are Inequalities?

An inequality refers to the comparison of two expressions and represents that one is greater than, less than, greater than or equal to, or less than or equal to the other.

Inequality Symbols:

> : Greater than

< : Less than

≥ : Greater than or equal to

≤ : Less than or equal to

Solving Inequalities

Though there is a basic difference when multiply or divide both sides by a negative number—check and reverse the inequality sign. Solving inequalities is like solving equations. Example: Solve 2x − 5 < 9. 1.Add 5 to both sides: 2x < 14. 2.Divide both sides by 2: x < 7. 3.Solution: All real numbers less than 7.

Special Rule: multiplying or dividing by negative numbers Example: Solve −3x > 9. 1.Divide both sides by −3 and reverse the inequality sign: x < −3. 2.Solution: All real numbers less than −3.

Algebra in the Real World

Solving Word Problems Translating real-world situations into algebraic expressions or equations allows us to solve problems efficiently. Example:

Problem:

A movie theatre charges ​10 for adults and ​8. If the theatre sells 250 tickets for a total of ​1,050, what is the number of adult tickets sold?

Solution:

1.Let a be the number of adult tickets and c the number of child tickets. 2.Set up equations:

Total tickets: a + c = 250.

Total sales: 10a + 8c = 1,050. 3.Solve the system using substitution or elimination. Simplify the solution. Use the Algebra Calculator to solve algebraic equations. Make Math Easier The Algebra Calculator is an adaptable online tool designed to simplify algebraic problem- compatible to solve the problems for all levels of users. Here's how to make the most of it:

Begin by entering the algebraic expression into the above input field or upload the image of the problem.

After entering the equation, click the 'Go' button to generate instant solutions.

The calculator provides detailed step-by-step solutions, aiding in understanding the underlying concepts.

How to Use an Algebra Calculator

Enter Your Problem: Type in your equation, expression, or system into the calculator's input field.

Select the operation: Choose the function you need: solve, simplify, factor, graph, etc.

Click Calculate: The calculator processes your input and provides a detailed solution.

Review the Steps: The step-by-step explanation helps you understand the process and learn how to solve similar problems.

Example:

Problem: Solve 5x - 6 = 3x - 8.

Calculator Solution:

Move 6 to the right side—> 5x = 3x - 2

Move 3x to the left side—> 2x = -2

Divide both sides by -2—> x = -1

Benefits of Using an Algebra Calculator

Saves Time: Resolves complex problems. Enhances Learning: Steps that are specific help people understand. Accessible Anywhere: Use it on any device with internet access. Boosts Confidence: Check your work and work on your problem-solving skills.

Conclusion

Mastering algebra simplifies the world, despite its initial appearance as a confusing network of symbols and equations. Algebra is the language of describing the workings of everything from financial calculations to engineering marvels. Gain access to limitless opportunities by developing strong analytical skills through regular practice, mastery of the fundamentals, and the use of useful tools such as our Algebra Calculator. Remember, even the most experienced professionals began their journey much later. Discover the captivating realm of algebra by embracing its challenges, remaining persistent, and savoring the ride!

Frequently Asked Questions (FAQ)

How do you solve algebraic expressions?

To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.

What are the basics of algebra?

The basics of algebra are the commutative, associative, and distributive laws.

What are the 3 rules of algebra?

The basic rules of algebra are the commutative, associative, and distributive laws.

What is the golden rule of algebra?

The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.

What are the 5 basic laws of algebra?

The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.