Linear Equations • Topic 2 of 3

Solving Linear Equations

What are Fractional Equations? Fractional equations contain fractions with the variable in the numerator, denominator, or both. To solve them, we eliminate denominators by multiplying both sides by the Least Common Multiple (LCM) of all denominators.

Steps to Solve Fractional Equations:

Find the LCM of all denominators
Multiply both sides of the equation by the LCM
Simplify (cancel denominators)
Solve the resulting linear equation
Check that the solution doesn't make any denominator zero

How to Approach Word Problems:

Step	Action
1	Read the problem carefully
2	Identify what is being asked (the unknown)
3	Choose a variable to represent the unknown
4	Translate the words into a mathematical equation
5	Solve the equation
6	Answer the question (with appropriate units)

Common Word Problem Types:

Age problems: Compare ages now and in the future
Number problems: Find unknown numbers based on given relationships
Money problems: Total cost, change, etc.
Consecutive integers: $n$, $n+1$, $n+2$
Perimeter/Geometry: Use formulas to set up equations

Solved Examples

Worked Example

Solve: $\frac{x+2}{3} = \frac{x-1}{2}$

Solution- LCM of 3 and 2 is 6 - Multiply both sides by 6: $6 \times \frac{x+2}{3} = 6 \times \frac{x-1}{2}$ - Simplify: $2(x+2) = 3(x-1)$ - Expand: $2x + 4 = 3x - 3$ - Subtract $2x$ from both sides: $4 = x - 3$ - Add 3 to both sides: $x = 7$ - Check: LHS = $(7+2)/3 = 9/3=3$, RHS = $(7-1)/2 = 6/2=3$ ✓ - **Answer:** $x = 7$ *Example 2 (Word Problem - Number): The sum of three consecutive integers is 72. Find the integers. Solution: - Let the first integer be $n$ - Then the three consecutive integers are $n$, $n+1$, $n+2$ - Sum: $n + (n+1) + (n+2) = 72$ - $3n + 3 = 72$ - $3n = 69$ - $n = 23$ - Integers: $23, 24, 25$ - Check: $23+24+25=72$ ✓ - **Answer:** $23, 24, 25$ *Example 3 (Word Problem - Age): A father is three times as old as his son. In 10 years, he will be twice as old as his son. Find their present ages. Solution: - Let son's present age = $x$ years - Father's present age = $3x$ years - In 10 years: son = $x + 10$, father = $3x + 10$ - Equation: $3x + 10 = 2(x + 10)$ - $3x + 10 = 2x + 20$ - $3x - 2x = 20 - 10$ - $x = 10$ (son's age) - Father's age = $3 \times 10 = 30$ - Check: In 10 years: son=20, father=40 (twice) ✓ - **Answer:** Son is 10 years, Father is 30 years

Key Points

To solve fractional equations, multiply by LCM of denominators
Always check that solution doesn't make any denominator zero
Word problems: read carefully, define variable, translate to equation
Consecutive integers: use $n$, $n+1$, $n+2$, etc.
Age problems: express future/past ages by adding/subtracting years
Always verify your answer in the original word problem
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.When a term is transposed across the $=$ sign, its:

Explanation: Sign changes.

Q2.To solve $2x=8$, you:

Explanation: Divide by $2$.

Q3.The solution of $3x-5=10$ is:

Explanation: $x=5$.

Q4.The solution of $\tfrac{x}{3}=4$ is:

Explanation: $x=12$.

Practice more MCQs → 📝 Assignment · 35 marks