Linear Equations in One Variable
Linear Equations in One Variable
Linear equations give 2 questions directly in the Part-III maths paper, and they underpin most of the algebraic word problems as well. The solving method itself is short. The marks are decided by how accurately you translate a worded situation into an equation before you solve it.
What Makes an Equation Linear
A linear equation in one variable has the variable to the power one only, and exactly one variable. It has exactly one solution.
ax + b = 0, where a ≠ 0, gives x = −b/a
- 2x + 5 = 11 is linear; x² + 3 = 7 is not, because the power is two.
- An equation with the variable on both sides is still linear; collect the terms first.
Solving Step by Step
- Clear brackets and fractions by multiplying through by the LCM of the denominators.
- Bring variable terms to one side and constants to the other, changing sign on transposition.
- Combine like terms, then divide by the coefficient of the variable.
- Substitute your answer back into the original equation to verify.
Solve 3(x − 2) = 2(x + 4): expand to 3x − 6 = 2x + 8, transpose to x = 14. Check: 3(12) = 36 and 2(18) = 36.
✗ Moving 5 across gives 2x = 11 + 5 in 2x + 5 = 11 | ✓ Transposing changes the sign: 2x = 11 − 5 = 6, so x = 3
Translating Word Problems
Almost every word problem uses one of a few patterns. Learn the translation, not the problem.
| Phrase | Translates to |
|---|---|
| A number is increased by 7 | x + 7 |
| 7 less than a number | x − 7 |
| Twice a number, decreased by 5 | 2x − 5 |
| Two consecutive integers | x and x + 1 |
| Two consecutive even or odd integers | x and x + 2 |
| Present age; age 5 years ago | x and x − 5 |
| Ratio 3 : 4 | 3k and 4k |
The 3k and 4k device is the single most useful trick here. Turning a ratio into a single unknown converts a two-variable problem into a linear equation you can actually solve.
Example: two numbers are in the ratio 3 : 4 and their sum is 63. Write 3k + 4k = 63, so 7k = 63 and k = 9. The numbers are 27 and 36.
Age and Digit Problems
Age problems need one fixed rule: the difference between two ages never changes, though the ratio does. If a father is three times his son's age now and will be twice in ten years, write 3x and x now, then 3x + 10 = 2(x + 10), giving x = 10.
For a two-digit number with tens digit a and units digit b, the number is 10a + b, and reversing it gives 10b + a. Their difference is always a multiple of 9 — a fact asked as a standalone question.
पक्षांतरण में चिह्न बदलता है; दो आयु का अंतर कभी नहीं बदलता।
TRE pointer: In a teaching paper the equation itself is rarely hard; the translation is. Expect at least one age or ratio word problem, and remember the age difference stays constant rule, which resolves most of them in one line. The digit-reversal facts — difference divisible by 9, sum divisible by 11 — are asked directly and are quick marks. Because every answer can be verified by substitution, this is a topic where you should never leave a question to option E.
60-Second Recap
- Linear means the variable appears to the power one only, with one solution.
- Clear brackets and fractions first; transposing a term changes its sign.
- Always substitute the answer back to verify.
- Ratio 3 : 4 becomes 3k and 4k — one unknown instead of two.
- The difference between two ages never changes, though the ratio does.
- A two-digit number is 10a + b; reversal differences are multiples of 9.
Frequently Asked Questions
How do I turn a ratio into an equation?
Introduce a single unknown multiplier. A ratio of three to four becomes 3k and 4k, so a problem with two quantities collapses into one linear equation. Solve for k and multiply back. This device handles most ratio word problems in a single line.
Why does the difference between two ages stay the same?
Because both people age at the same rate, so the gap between them never changes even though the ratio of their ages does. This is why age problems are set up with the ratio changing over time while the difference stays fixed, and that fixed difference solves them.
How do I represent a two-digit number in algebra?
If the tens digit is a and the units digit is b, the number is ten a plus b, and the number formed by reversing the digits is ten b plus a. Their difference is always a multiple of nine and their sum is always a multiple of eleven.
What happens to a sign when I move a term across the equals sign?
It changes. Adding on one side becomes subtracting on the other, and multiplying becomes dividing. This is called transposition and it is simply doing the same operation to both sides. Forgetting the sign change is the most frequent slip in solving linear equations.