Algebra & Identities • Topic 3 of 4

Linear Equations

A linear equation in one variable is solved by isolating it; two equations in two variables are solved by substitution or elimination. SSC word problems (ages, coins, two-digit numbers) translate one statement into one equation. For a two-digit number with tens t and units u, its value is 10t + u and reversing gives 10u + t.

✅ Solved examples

1. Solve 3x + 5 = 20.

3x = 15 -> x = 5.

2. Solve 2x + 3y = 12 and x - y = 1.

From second x = y + 1; substitute: 2(y+1)+3y = 12 -> 5y = 10 -> y = 2, x = 3.

3. The sum of two numbers is 30 and their difference is 8. Find them.

Numbers (30+8)/2 = 19 and (30-8)/2 = 11.

4. The sum of the digits of a two-digit number is 9, and the number is 9 more than the number formed by reversing its digits. Find it.

Let tens t, units u. t + u = 9 and (10t+u) - (10u+t) = 9 -> 9(t-u) = 9 -> t - u = 1. Solving: t = 5, u = 4. Number = 54.

✏️ Practice — try these, take hints as needed

1. Solve 5x - 7 = 18.

5x = 25.

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2. Solve x + y = 10, x - y = 4.

Add equations.

2x = 14.

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x = 7, y = 3

3. Sum 40, difference 6. Numbers?

(40+6)/2 and (40-6)/2.

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23 and 17

4. Solve 4x + 2 = 2x + 10.

2x = 8.

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5. 3x + 2y = 16 and x = 2. Find y.

6 + 2y = 16.

2y = 10.

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📝 Topic test — 8 questions

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Formula Reference Sheet

This chapter

Identities

Square of sum	(a + b)^2 = a^2 + 2ab + b^2
Difference of squares	a^2 - b^2 = (a + b)(a - b)
Cube of sum	(a + b)^3 = a^3 + b^3 + 3ab(a + b)
Sum/diff of cubes	a^3 +/- b^3 = (a +/- b)(a^2 -/+ ab + b^2)
Three-term	a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

x + 1/x family

Square	x^2 + 1/x^2 = (x + 1/x)^2 - 2
Cube	x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x)
Quadratic roots	x = (-b +/- root(b^2 - 4ac)) / 2a

SSC reference