Exercise · ZW-SYSTEM-OF-LINEAR-EQUATIONS-010 · ★★★ Advanced
About a homogeneous system (all constant terms equal to 0), which statement is always true?
Choose one option.
Show answer
- C. It has at least the trivial solution (all unknowns equal to 0)
Explanation
Substituting 0 for every unknown satisfies each equation, so the trivial solution always exists. That is why a homogeneous system is never inconsistent; it has either a unique solution or infinitely many, depending on the case.
Review the concept: System of linear equations
A system of linear equations is a set of linear equations in the same unknowns. A solution is an assignment of values that satisfies all the equations at once.
\[\begin{aligned}a_{11}x_1+\cdots+a_{1n}x_n&=b_1\\ \vdots\qquad&\\ a_{m1}x_1+\cdots+a_{mn}x_n&=b_m\end{aligned}\]
More practice problems: system of linear equations
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-001 Which of the following equations is linear in the unknowns x and y?
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-002 Solve the system x+y=5,\; x-y=1.
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-003 Which of the following systems has (x,y)=(2,1) as a solution?
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-004 How many solutions does the system x+y=2,\; 2x+2y=4 have?
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-005 How many solutions does the system x+y=1,\; x+y=3 have?
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-006 For which value of k does the system x+y=1,\; 2x+2y=k have infinitely many solutions?
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-007 Solve the system x+y+z=6,\; x-y+z=2,\; 2x+y-z=1.
- ZW-SYSTEM-OF-LINEAR-EQUATIONS-008 What is the augmented matrix of the system 2x-y=3,\; x+4y=-1?