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MATH / There exists a real number k such that the equation

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problem

There exists a real number kk such that the equation \[\begin{pmatrix} 4 \\ -1 \end{pmatrix} + t \begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} 8 \\ k \end{pmatrix} + s \begin{pmatrix} -15 \\ -6 \end{pmatrix}\]has infinitely many solutions in tt and ss. Find kk.
Plain-text mathematical notation (without MathML)
There exists a real number k such that the equation
\[\begin{pmatrix} 4 \\ -1 \end{pmatrix} + t \begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} 8 \\ k \end{pmatrix} + s \begin{pmatrix} -15 \\ -6 \end{pmatrix}\]has infinitely many solutions in t and s.  Find k.
Original LaTeX notation
There exists a real number $k$ such that the equation
\[\begin{pmatrix} 4 \\ -1 \end{pmatrix} + t \begin{pmatrix} 5 \\ 2 \end{pmatrix} = \begin{pmatrix} 8 \\ k \end{pmatrix} + s \begin{pmatrix} -15 \\ -6 \end{pmatrix}\]has infinitely many solutions in $t$ and $s$.  Find $k$.

level

Level 3

type

Precalculus

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Source and history

task_id
4fa0a07f-c779-5980-a029-34d4e3baf467
task_key
precalculus--test--4fa0a07f-c779-5980-a029-34d4e3baf467
task_revision_id
1
config / split
precalculus / test

Official source

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