If you have worked steadily through the Programme, you...

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward. Find the values of $k$ for which the following equations are consistent: $$ \left\{\begin{array}{c} 3 x+5 y+k=0 \\ 2 x+y-5=0 \\ (k+1) x+2 y-10=0 \end{array}\right. $$

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward.
Find the values of $k$ for which the following equations are consistent:
$$
\left\{\begin{array}{c}
3 x+5 y+k=0 \\
2 x+y-5=0 \\
(k+1) x+2 y-10=0
\end{array}\right.
$$

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Key Concepts

System of Linear Equations

These are collections of two or more linear equations involving the same set of variables. Solving these systems is a central topic in mathematics and engineering, often requiring the use of techniques such as substitution, elimination, or matrix methods to find the solution sets.

Consistency of Linear Systems

The concept of consistency in a system of linear equations refers to whether the system has at least one solution. A consistent system is one in which all the equations can be satisfied simultaneously, while an inconsistent system leads to a contradiction, meaning there are no solutions.

Parameter Dependence in Equations

Introducing a parameter into a system of equations allows the study of how the system's behavior changes with different parameter values. Determining the values of the parameter that result in a consistent or unique solution requires careful manipulation and analysis of the relationships between the equations.

Rank and the Rouché–Capelli Theorem

This theorem provides a criterion for the consistency of a linear system. It states that a system of equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. This concept is fundamental when analyzing systems that include parameters or potential redundancies.

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