# For which values of the constants c and d is begin{bmatrix}5 7c d end{bmatrix} a linear combination of begin{bmatrix}1 11 1 end{bmatrix} text{ and } begin{bmatrix}1 23 4 end{bmatrix}

For which values of the constants c and d is $\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]$ a linear combination of
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Step 1 Calculation: Write the vector $\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]$ in linear combination of
$⇒\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]=A\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]+B\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$
So we get four linear equations in A and B
$⇒\left\{\begin{array}{l}A+B=5\\ A+2B=7\\ A+3B=c\\ A+4B=d\end{array}$
consider first two linear equations $⇒\left\{\begin{array}{l}A+B=5\\ A+2B=7\end{array}$
By solving these equations we get A=3 , B=2
Now we have equations $\left\{\begin{array}{l}A+3B=c\\ A+4B=d\end{array}$
Substituting values A=3 , B=2 in above system we get $\left\{\begin{array}{l}c=3+3\left(2\right)=9\\ d=3+4\left(2\right)=11\end{array}$
$⇒c=9,d=11$
Hence for c=9 , d=11 we can write the vector $\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]$ in linear combination of Step 2 Answer: Hence for c=9 , d=11 we can write the vector $\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]$ in linear combination
Jeffrey Jordon