Mathematics

Mathematics

Problem

The simultaneous equations px+y=0px + y = 0 and 4x+(p+3)y=04x + (p+3)y = 0 have infinite solutions for A. p=−4,1p = -4, 1 B. p=−1,4p = -1, 4 C. p=−4,−1p = -4, -1 D. p∈R∖{−4,1}p \in \mathbb{R} \setminus \{-4, 1\} E. p∈R∖{−1,4}p \in \mathbb{R} \setminus \{-1, 4\} A. p=−4,1p = -4, 1 B. p=−1,4p = -1, 4 C. p=−4,−1p = -4, -1 D. p∈R∖{−4,1}p \in \mathbb{R} \setminus \{-4, 1\} E. p∈R∖{−1,4}p \in \mathbb{R} \setminus \{-1, 4\}
Intuitive explanation

A homogeneous system of linear equations has infinite solutions when the equations are proportional—that is, one is a scalar multiple of the other. This means the coefficient matrix must be singular, so its determinant equals zero. Set up the determinant of the coefficient matrix, solve for pp, and you'll find the values that make the system dependent rather than independent.

Intuitive explanation

A homogeneous system of linear equations has infinite solutions when the equations are proportional—that is, one is a scalar multiple of the other. This means the coefficient matrix must be singular, so its determinant equals zero. Set up the determinant of the coefficient matrix, solve for pp, and you'll find the values that make the system dependent rather than independent.

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