Exercise · ZW-SYSTEM-OF-LINEAR-EQUATIONS-009 · ★★★ Advanced
For which values of \(k\) does the system \(x+2y=1,\; 3x+ky=2\) have a unique solution?
Choose one option.
Show answer
- B. \(k\neq 6\)
Explanation
The system has a unique solution when the lines are not parallel, that is, when \(1\cdot k-2\cdot 3\neq 0\), so \(k\neq 6\). If \(k=6\) the lines are parallel (and distinct, since \(3\cdot 1\neq 2\)): there is no solution.
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?
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