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For every value of a, the graph of each of these three equations is a line:

ax + y = 1

x  + ay = 1

x + y = 9

 

For what value of a do all three of these lines intersect at the same point?

 Jul 6, 2021
 #1
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For every value of a, the graph of each of these three equations is a line:
\(ax + y = 1 \\ x + ay = 1\\ x + y = 9\)
For what value of a do all three of these lines intersect at the same point?

 

My attempt:

\(\begin{array}{|lrcll|} \hline (1): & ax + y &=& 1 \\ (2): & x + ay &=& 1 \\ (3): & x + y &=& 9 \\ \hline \end{array}\)

 

\(\begin{array}{|lrcll|} \hline (1): & ax + y &=& 1 \\ & y &=& 1-ax \\\\ (3): & x + y &=& 9 \\ & x+1-ax &=& 9 \\ & x(1-a) &=& 8 \\ & \mathbf{x} &=& \mathbf{\dfrac{8}{1-a}} \\\\ & y &=& 9-x \\ & \mathbf{y} &=& \mathbf{9-\dfrac{8}{1-a}} \qquad (4) \\\\ \hline \end{array}\)

 

\(\begin{array}{|lrcll|} \hline (3): & x + y &=& 9 \\ & x &=& 9-y \\\\ (2): & x + ay &=& 1 \\ & 9-y + ay &=& 1 \\ & y(1-a) &=& 8 \\ & \mathbf{y} &=& \mathbf{\dfrac{8}{1-a}} \qquad (5) \\ \hline \end{array}\)

 

\(\begin{array}{|lrcll|} \hline (5)=(4):& y = \dfrac{8}{1-a} &=& 9-\dfrac{8}{1-a} \\\\ & \dfrac{8}{1-a} &=& 9-\dfrac{8}{1-a} \\\\ & \dfrac{2*8}{1-a} &=& 9 \\\\ & \dfrac{16}{1-a} &=& 9 \\\\ & 16 &=& 9(1-a) \\ & 16 &=& 9-9a \\ & 9a &=& 9-16 \\ & 9a &=& -7 \\\\ & \mathbf{a} &=& \mathbf{ -\dfrac{7}{9} } \\ \hline \end{array}\)

 

laugh

 Jul 6, 2021

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