📐
➗
📊
✖️
➕
← Back to Dashboard
Font Size:

1. Introduction

We come across many situations in everyday life where we need to find an unknown quantity. For example, we may want to know the price of a pen when we know the total cost of a few pens, or the number of marbles in a bag when the total count and the shared number are known. In mathematics, such unknown quantities are represented by variables (usually letters like x, y or z), and the relationship between the known and unknown quantities is written as an equation. Solving that equation gives us the value of the unknown.

An algebraic equation is a statement of equality between two algebraic expressions. A linear equation in one variable is an equation of the form ax + b = 0, where a and b are constants and a is not zero. For example, 2x + 3 = 7 is a linear equation in one variable x. In this chapter we will learn how to solve such equations, how to use transposition and cross multiplication, how to reduce equations to simpler forms, and how to apply linear equations to solve real-life word problems.

2. Understanding Equations and Their Solutions

An equation has two sides: the left-hand side (LHS) and the right-hand side (RHS), connected by the equality sign. The value of the variable which satisfies the equation, that is, which makes LHS equal to RHS, is called the solution of the equation.

For example, in the equation 2x + 3 = 7, putting x = 2 gives LHS = 2(2) + 3 = 7 = RHS. So x = 2 is the solution of the equation.

Balanced Equation Idea

An equation is like a weighing balance which is balanced. Whatever operation we perform on one side must be performed on the other side as well, so that the balance is not disturbed.

3. Solving Linear Equations

Transposition Method

The process of moving a term from one side of the equation to the other side with its sign changed is called transposition.

Example: Solve 2x + 3 = 7. Transpose 3 to RHS: 2x = 7 - 3 = 4. Transpose 2 to RHS: x = 4/2 = 2.

Solving Equations with Variables on Both Sides

When variables appear on both sides, first bring all variable terms to one side and all constant terms to the other side, then solve.

Example: Solve 2x - 3 = x + 5. 2x - x = 5 + 3, so x = 8.

4. Reducing Equations to Simpler Form

Equations with Fractions (Cross Multiplication)

To solve equations containing fractions, we multiply both sides by the LCM of the denominators, or use cross multiplication.

For example, to solve (x + 1)/2 = (x - 3)/4, we cross multiply: 4(x + 1) = 2(x - 3). This gives 4x + 4 = 2x - 6, so 4x - 2x = -6 - 4, that is, 2x = -10, hence x = -5.

Equations Reducible to Linear Form

Some equations do not look linear but can be reduced to linear form by cross multiplication. For example, (2x + 1)/(3x - 2) = 5/9 can be cross multiplied to get 9(2x + 1) = 5(3x - 2), which is a linear equation.

5. Applications of Linear Equations in Word Problems

To solve word problems we follow these steps: 1. Read the problem carefully and identify the unknown quantity. Let the unknown be x. 2. Form a linear equation using the given conditions. 3. Solve the equation. 4. Verify the answer by substituting it back in the problem.

Example

The sum of three consecutive integers is 51. Find the integers. Let the integers be x, x + 1 and x + 2. x + (x + 1) + (x + 2) = 51 3x + 3 = 51, so 3x = 48 and x = 16. The integers are 16, 17 and 18.

Quick Revision Tables

Table 1: Transposition Rules

Operation on LHS Operation on RHS after transposition
Addition (+a) Subtraction (-a)
Subtraction (-a) Addition (+a)
Multiplication (x a) Division (divided by a)
Division (divided by a) Multiplication (x a)

Table 2: Steps to Solve Word Problems

Step Action
Step 1 Read the problem and identify the unknown, let it be x
Step 2 Form the equation using given conditions
Step 3 Solve the equation using transposition
Step 4 Check the solution by substitution
Step 5 Write the answer with proper units

Mind Map

graph TD A["Linear Equations in One Variable"] --> B["Form: ax + b = 0, a not equal to 0"] A --> C["Solution: value of variable making LHS = RHS"] A --> D["Methods"] D --> E["Transposition: move term, change sign"] D --> F["Cross multiplication for fractions"] D --> G["Bring variable terms to one side"] A --> H["Reducible forms"] H --> I["Multiply both sides by LCM of denominators"] A --> J["Word Problems"] J --> K["Let unknown = x, form equation, solve, verify"]

Important Diagrams (SVG)

Diagram 1: The Balancing Act of an Equation

An Equation is a Balanced Scale LHS 2x + 3 RHS 7 Whatever you do to one side, do the same to the other. Subtract 3 from both sides, then divide both sides by 2. Golden Rule: Perform the same operation on both sides to keep the equation balanced.

Diagram 2: Steps to Solve a Linear Equation

Solving 3(x + 2) = 15 Step 1: Expand brackets 3x + 6 = 15 Step 2: Transpose constant 3x = 15 - 6 = 9 Step 3: Transpose coefficient x = 9/3 = 3 Step 4: Verify 3(3 + 2) = 15, correct Tip: When terms have variables on both sides, bring them together first. Cross multiply to remove fractions. Golden Rule: Isolate the variable; always verify your solution by substitution.

Common Mistakes

  1. Students often forget to change the sign while transposing a term. For example, in x + 4 = 9, moving 4 to the other side must give x = 9 - 4, not x = 9 + 4.
  2. While removing brackets, students forget to multiply the number outside with every term inside the brackets. (x + 2) x 3 is 3x + 6, not x + 6.
  3. When cross multiplying fractions, students sometimes multiply incorrectly. (x + 1)/2 = (x - 3)/4 gives 4(x + 1) = 2(x - 3), not x + 1 = 2(x - 3).
  4. Students forget to check their answer by substituting it back into the original equation, which is essential to catch errors.
  5. In word problems, students often get the sign wrong when forming the equation, such as using "sum" as difference or forgetting that consecutive integers differ by 1.
  6. When the same variable appears on both sides, students fail to combine the like terms correctly. For example 2x - x is x, not 0.
  7. Students forget that an equation of the form a/b = c/d can be solved by cross multiplication only when b and d are not zero.

Exam Tips

  1. Write "Let x be the unknown" at the start of every word problem to gain method marks.
  2. Show every transposition step clearly; do not skip steps, as stepwise marking is common in exams.
  3. In equations with fractions, first eliminate denominators by multiplying both sides by the LCM of all denominators.
  4. Always end with "Verification: substituting x = ... in the equation, LHS = RHS" to score full marks.
  5. Practise cross multiplication, since it appears in almost every test on this chapter.
  6. For word problems, always attach units to the final answer, such as rupees, years, or centimetres.
  7. Revise by solving one equation and one word problem every day; speed and accuracy both matter in the exam hall.

Conclusion

Linear equations in one variable are one of the most important tools of algebra. In this chapter we learnt what an equation is, how to find its solution, and how to use transposition and cross multiplication to solve even those equations which appear complicated or fractional. We also studied how to convert real-life situations into equations and solve them, which is a skill used throughout mathematics and science. The concepts of this chapter are the foundation for simultaneous equations, quadratic equations and inequalities that will be studied in higher classes. Mastering the balance idea and practising many word problems is the key to scoring well in examinations.