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.

Coincident Equations With Infinitely Many Solutions

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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Parallel Equations With No Solution

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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Substitution With an Isolated Variable

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

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Elimination After Scaling

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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Elimination With Fractional Coefficients

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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Three-Variable 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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Cramer's Rule With a Nonzero Determinant

Solve

2x+y=7,x−y=−12x+y=7,\qquad x-y=-1

using Cramer's Rule, then check the result.

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Singular Three-Variable System With a Free Parameter

Solve and classify

x+y+z=4,2x+2y+2z=8,x−y=0.x+y+z=4,\qquad 2x+2y+2z=8,\qquad x-y=0.

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