Linear Algebra and Its Applications · Problem 1.2.17

Matrices and Gaussian Elimination — Gilbert Strang

Mathematics · geometry-of-linear-equations

Problem

The first of these equations plus the second equals the third: x+y+z=2x+2y+z=32x+3y+2z=5.\begin{aligned} x + y + z &= 2 \\ x + 2y + z &= 3 \\ 2x + 3y + 2z &= 5. \end{aligned} The first two planes meet along a line. The third plane contains that line, because if xx, yy, zz satisfy the first two equations then they also \_\_\_\_\_\_. The equations have infinitely many solutions (the whole line L\mathbf{L}). Find three solutions.
Intuitive explanation

The key insight is recognizing that the third equation is literally a linear combination of the first two—it's their sum. This means any point satisfying both the first and second equations automatically satisfies the third, so all three planes share the same line of intersection. To find solutions, treat one variable as a free parameter, substitute into the first two equations to solve for the other two, and generate points by varying that parameter.

Intuitive explanation

The key insight is recognizing that the third equation is literally a linear combination of the first two—it's their sum. This means any point satisfying both the first and second equations automatically satisfies the third, so all three planes share the same line of intersection. To find solutions, treat one variable as a free parameter, substitute into the first two equations to solve for the other two, and generate points by varying that parameter.

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