Solving Linear Equations Variable On One Side

Unit: Linear Equation with one Variable

Chapter: Solving Linear Equations Variable on one side

Reference: – Definition of a Linear Equation in One Variable, Formation of Linear Equations, Properties of Equality in Solving Equations, Solving Linear Equations with the Variable on One Side, Simplification Techniques for Linear Equations, Checking Solutions of Linear Equations, Word Problems Involving Linear Equations

After studying this chapter, you should be able to understand:

  • Definition of a Linear Equation in One Variable
  • Properties of Equality in Solving Equations
  • Simplification Techniques for Linear Equations
  • Checking Solutions of Linear Equations
  1. Definition of a Linear Equation in One Variable
    • A linear equation in one variable is an algebraic equation that consists of a single unknown quantity and follows a specific mathematical structure.
    • The equation involves terms that include the variable raised only to the first power, ensuring a straight-line relationship when graphed.
    • It represents a fundamental concept in algebra that helps in solving for an unknown value based on given conditions.
  2. Formation of Linear Equations
    • Linear equations can be formed by translating real-world situations or algebraic expressions into mathematical statements.
    • The construction of these equations often involves recognizing relationships between quantities and expressing them symbolically.
    • Understanding how to frame linear equations correctly is crucial for solving problems in various practical applications.
  3. Properties of Equality in Solving Equations
    • The properties of equality state that any operation applied to one side of an equation must also be applied to the other side to maintain balance.
    • These properties include addition, subtraction, multiplication, and division, which help simplify the equation systematically.
    • By applying these properties correctly, the unknown variable can be isolated to determine its value.
  4. Solving Linear Equations with the Variable on One Side
    • A linear equation with a variable on one side requires rearranging terms to express the unknown quantity explicitly.
    • This process involves simplifying expressions, moving constants to the opposite side, and systematically eliminating coefficients.
    • The goal is to isolate the variable and determine its exact value while maintaining the equality of the equation.
  5. Simplification Techniques for Linear Equations
    • Solving equations often requires simplification techniques such as combining like terms and eliminating fractions or decimals.
    • These methods ensure that the equation is expressed in its simplest form, making it easier to solve.
    • Simplification is essential for reducing complexity and improving accuracy in solving algebraic equations.
  6. Checking Solutions of Linear Equations
    • Once a solution is obtained, it is necessary to verify its correctness by substituting it back into the original equation.
    • Checking ensures that the solution satisfies the equation and maintains equality.
    • This validation step helps prevent errors and confirms that the solution is accurate.
  7. Word Problems Involving Linear Equations
    • Many real-world problems can be expressed as linear equations, making it essential to understand how to translate verbal descriptions into mathematical statements.
    • The process involves identifying unknowns, setting up an equation based on given conditions, and solving for the required quantity.
    • Mastering this skill enhances problem-solving abilities and provides a practical application of algebra in everyday life.

Example: –

A bookstore sells a combination of fiction and non-fiction books. The cost of a fiction book is $x, while the cost of a non-fiction book is $2x – 5. A customer buys 7 fiction books and 5 non-fiction books and pays a total of $275.

Additionally, if the cost of each fiction book was increased by $3, and the cost of each non-fiction book remained the same, the total price for the same number of books would have been $296.

Using the concepts of linear equations, properties of equality, simplification techniques, and verification, determine:

  1. The cost of a fiction book and a non-fiction book.
  2. Verify the solution by substituting it back into both conditions.

Solution: –
 

Step 1: Forming the First Equation

Total cost of 7 fiction books and 5 non-fiction books is $275:

7(x)+5(2x−5)=275

Expanding the equation:

7x+10x−25=275
17x−25=275

Adding 25 to both sides:

17x=300

Dividing by 17:

Thus, the cost of a fiction book is $17.65.

For the non-fiction book:

2x−5=2(17.65) −5=35.3−5=30.3

Thus, the cost of a non-fiction book is $30.30.

 

Step 2: Forming the Second Equation

If the price of fiction books was increased by $3, the new price of a fiction book becomes:

x+3=17.65+3=20.65

The total price for 7 fiction books and 5 non-fiction books is now $296:

Since the calculated value is very close to the actual given value (296$)

Conclusive Points for Introduction to Linear Equations & Solving Linear Equations with Variable on One Side

  1. Linear Equations Provide a Foundation for Algebra
    • Understanding linear equations in one variable is essential for solving more complex algebraic problems.
    • Mastering this concept helps in developing logical thinking and problem-solving skills in mathematics.
  2. Balance is Key in Solving Equations
    • Any operation performed on one side of an equation must also be applied to the other side to maintain equality.
    • This principle ensures that the integrity of the equation remains unchanged throughout the solving process.
  3. Systematic Simplification Leads to Correct Solutions
    • Combining like terms, eliminating fractions, and rearranging expressions in a structured manner simplifies the equation.
    • A step-by-step approach reduces errors and makes solving linear equations more efficient.
  4. Verification Confirms the Accuracy of Solutions
    • Substituting the obtained value back into the original equation helps verify correctness.
    • This step is crucial to ensure that the solution satisfies the given equation.
  5. Real-World Applications Make Linear Equations Relevant
    • Linear equations are widely used in financial calculations, business planning, physics, and everyday problem-solving.
    • Learning to form and solve these equations enhances analytical skills and practical mathematical understanding.

 

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