Class 8 — Mathematics (NCERT)
Linear Equations in One Variable
Class 8
- ✓By the end of this lesson students will be able to define and identify linear equations in one variable.
- ✓By the end of this lesson students will be able to solve linear equations with variables on one side and on both sides.
- ✓By the end of this lesson students will be able to formulate and solve word problems involving linear equations in one variable.
- ✓By the end of this lesson students will be able to reduce equations to linear form and solve them.
Key concepts
An algebraic equation is an equality involving variables and constants. A linear equation in one variable is an equation that can be written in the form ax + b = 0, where 'a' and 'b' are real numbers and 'a ≠ 0', and 'x' is the variable. The highest power (exponent) of the variable in such an equation is 1. For example, 2x + 5 = 0, y - 7 = 3, etc.
Solving an equation means finding the value of the variable that makes the equation true. This value is called the solution or root of the equation. We solve an equation by performing the same mathematical operations (addition, subtraction, multiplication, division) on both sides of the equation to isolate the variable on one side.
The transposition method is a convenient way to solve equations. When a term is moved (transposed) from one side of the equation to the other, its sign changes. For example, a term added on one side becomes subtracted on the other, and a term multiplied on one side becomes divided on the other. This helps in collecting like terms.
Some linear equations may have the variable appearing on both the Left Hand Side (L.H.S.) and the Right Hand Side (R.H.S.). To solve such equations, we first collect all terms containing the variable on one side (usually L.H.S.) and all constant terms on the other side (R.H.S.) using the transposition method. Then, we simplify and solve for the variable.
Some equations may not appear to be linear initially, especially those involving fractions where the variable is in the denominator. However, by performing suitable algebraic manipulations, such as cross-multiplication or multiplying both sides by the L.C.M. of the denominators, these equations can be converted into the standard linear form (ax + b = 0 or ax + b = cx + d), which can then be solved using the methods discussed.
Key facts to remember
- 1A linear equation in one variable has only one solution (root).
- 2The highest power of the variable in a linear equation is 1.
- 3Performing the same mathematical operation (addition, subtraction, multiplication, division by a non-zero number) on both sides of an equation maintains its equality.
- 4When transposing a term from one side of the equation to the other, its sign must be changed (e.g., '+' becomes '-', '×' becomes '÷').
- 5To verify a solution, substitute the value of the variable back into the original equation and check if L.H.S. = R.H.S.
- 6Word problems require careful reading and translation into correct algebraic equations before solving.
Worked examples
Example 1
Solve: 3x - 5 = 13
Answer
x = 6
To check the solution, substitute x = 6 in the original equation: L.H.S. = 3(6) - 5 = 18 - 5 = 13. R.H.S. = 13. Since L.H.S. = R.H.S., the solution is correct.
Example 2
Solve: 5x + 7/2 = 3/2 x - 14
Answer
x = -5
Always ensure to take the correct L.C.M. when dealing with fractional terms.
Example 3
The sum of two numbers is 75. One of the numbers is 15 more than the other. Find the numbers.
Answer
The two numbers are 30 and 45.
Check your answer: 30 + 45 = 75 (correct sum). 45 is 15 more than 30 (correct difference).
Example 4
Solve: (x + 1) / (2x + 3) = 3/8
Answer
x = 1/2
Remember that cross-multiplication is valid only when you have a single fraction on each side of the equality sign.
Common mistakes
- ✗Incorrectly changing the sign of a term when transposing it from one side of the equation to the other.
- ✗Making errors in basic arithmetic calculations, especially with negative numbers or fractions.
- ✗Not distributing negative signs correctly when removing brackets (e.g., -(x - 2) is -x + 2, not -x - 2).
- ✗Incorrectly forming the algebraic equation from a given word problem, leading to an incorrect solution.
- ✗Forgetting to perform the same operation on both sides of the equation, thus disturbing the equality.
- ✗Errors in cross-multiplication or finding the L.C.M. when reducing equations to linear form.
Exam tips
- ★Always show all steps of your working clearly and logically. This helps in identifying errors and earns partial marks even if the final answer is wrong.
- ★After finding a solution, always verify it by substituting the value of the variable back into the original equation to ensure L.H.S. = R.H.S.
- ★Read word problems carefully multiple times to understand the context, identify the unknown quantity, and correctly translate the problem into an algebraic equation.
- ★Practice a wide variety of problems, including those with fractions, decimals, and variables on both sides, to gain confidence and improve speed.
- ★Maintain neatness and organisation in your answer sheet to avoid silly calculation errors and make your work easy to follow for the examiner.
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