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🌟 The Illustrated Guide to Algebra

I. Linear Equations

Example: Solve \(2x + 3 = 7\)

Step 1: Subtract 3: \(2x = 4\)
Step 2: Divide by 2: \(x = 2\)

II. Quadratic Equations

Example: Solve \(x^2 - 4x + 3 = 0\)

Factorize: \((x - 1)(x - 3) = 0\)
Solutions: \(x = 1, x = 3\)

III. Polynomials

Example: Evaluate \(p(x) = x^3 - 2x^2 + x - 1\) at \(x = 2\)

\(p(2) = 2^3 - 2 \cdot 2^2 + 2 - 1 = 8 - 8 + 2 - 1 = 1\)

IV. Functions

Example: For \(f(x) = 2x^2 + 1\), find \(f(3)\)

\(f(3) = 2 \cdot 3^2 + 1 = 2 \cdot 9 + 1 = 19\)

V. Inequalities

Example: Solve \(3x - 5 > 4\)

Add 5: \(3x > 9\)
Divide by 3: \(x > 3\)

VI. Systems of Equations

Example: Solve \(\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}\)

Add equations: \(3x = 6 \to x = 2\)
Substitute: \(2 + y = 5 \to y = 3\)

VII. Sequences

Example: Arithmetic sequence with first term 2, common difference 3

General term: \(a_n = 2 + 3(n-1)\)
First 5 terms: \(2, 5, 8, 11, 14\)