This section provides step-by-step examples for solving systems of linear equations. You will learn how to apply the Substitution and Elimination methods to find the intersection points of lines in 2D and planes in 3D.

Case Study 1: Identifying a System with No Solution

Problem: Solve the following system and classify it: Equation 1: 2x−y=52x - y = 5 Equation 2: 4x−2y=104x - 2y = 10

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Case Study 2: Identifying an Inconsistent System

Problem: Solve the following system and classify it: Equation 1: 3x+y=43x + y = 4 Equation 2: 6x+2y=156x + 2y = 15

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Example 1: Solving by Substitution (Basic)

Problem: Solve the system of equations: Equation 1: x=3y−1x = 3y - 1 Equation 2: 2x+5y=202x + 5y = 20

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Example 2: Solving by Elimination (Intermediate)

Problem: Solve the system of equations: Equation 1: 4x−3y=254x - 3y = 25 Equation 2: −3x+8y=10-3x + 8y = 10

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Example 3: Fractional Coefficients (Advanced)

Problem: Solve the system of equations: Equation 1: 12x+23y=1\frac{1}{2}x + \frac{2}{3}y = 1 Equation 2: 34x−16y=2\frac{3}{4}x - \frac{1}{6}y = 2

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Example 4: $3 \times 3$ System by Elimination

Problem: Solve the system: Eq 1: x+2y−z=4x + 2y - z = 4 Eq 2: 2x−y+3z=92x - y + 3z = 9 Eq 3: −x+3y+2z=7-x + 3y + 2z = 7

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