Linear Equations in One Variable
Overview
Linear Equations in One Variable is a fundamental chapter in Class 8 Mathematics. It deals with equations that have exactly one variable and the highest power of that variable is one. Students learn to solve equations where variables appear on both sides, equations reducible to linear form, and real-life word problems. Understanding this chapter builds a strong algebraic foundation for higher classes and helps in logical problem-solving.
Typed Notes
Introduction to Linear Equations
A linear equation in one variable is an equation where the variable appears with power 1. For example, 2x + 5 = 11 is a linear equation. The solution is the value of the variable that makes the equation true. To solve, we perform the same mathematical operations on both sides to keep the equation balanced. We can add, subtract, multiply, or divide both sides by the same non-zero number without disturbing the equality.
Solving Equations with Variables on Both Sides
Sometimes variables appear on both sides of the equation, for example 3x + 4 = x + 10. To solve, collect all variable terms on one side and constant terms on the other side using transposition. Transposing means shifting a term to the other side with its sign reversed. Here: 3x − x = 10 − 4, so 2x = 6, giving x = 3. Always verify the answer by substituting back into the original equation to confirm both sides are equal.
Equations Reducible to Linear Form
Some equations look complex but can be reduced to linear form. For example, equations involving fractions like (x+1)/(x+2) = 3/4 can be solved using cross-multiplication. Cross-multiplying gives 4(x+1) = 3(x+2), which simplifies to 4x + 4 = 3x + 6, so x = 2. This technique converts a fractional equation into a simple linear equation. Always check that the denominator does not become zero for the solution obtained.
Word Problems Based on Linear Equations
Real-life situations can be translated into linear equations. Steps: (1) Read the problem carefully. (2) Identify the unknown and assign it a variable, say x. (3) Form an equation using the given conditions. (4) Solve the equation. (5) Verify the answer against the original problem. For example, if a number is doubled and 5 is added the result is 17, we write 2x + 5 = 17, solve to get x = 6. Word problems include age problems, number problems, and geometry problems.
Key Concepts
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- 01A linear equation in one variable has the form ax + b = 0, where a ≠ 0 and x is the variable.
- 02Transposing a term means moving it to the other side of the equation with its sign changed.
- 03An equation remains balanced if the same operation is performed on both sides.
- 04Cross-multiplication is used to solve equations of the form p/q = r/s, giving p×s = q×r.
- 05Word problems are solved by forming an equation from the given conditions and then solving for the unknown.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
Solve: 3x + 7 = 22
Solve: 5x − 3 = 2x + 9
Solve: (x + 2) / 3 = (x − 1) / 2
The sum of three consecutive integers is 51. Find the integers.
The present age of a father is three times the age of his son. After 10 years, father's age will be twice the son's age. Find their present ages.
Linear Equations in One Variable involve finding the value of an unknown that satisfies an equation with one variable raised to power one. Key methods include transposition, performing identical operations on both sides, and cross-multiplication for fractional equations. Word problems are solved by forming equations from given conditions. Verifying the solution by substituting back is always recommended to ensure correctness.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.What is the solution of 2x + 6 = 14?
Q2.Which of the following is a linear equation in one variable?
Q3.If (x/4) = 5, what is the value of x?
Q4.Transposing +8 from the left side to the right side changes it to:
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