1957 AHSME Problems/Problem 41
Problem
Given the system of equations
![\[ax + (a - 1)y = 1\]](http://latex.artofproblemsolving.com/6/7/c/67c81eabd802f9e2304407b799d8f4b7a8c6fd44.png)
![\[(a + 1)x - ay = 1\]](http://latex.artofproblemsolving.com/f/5/f/f5fa0d3f6f8278695383868623baf129c4d40616.png)
For which one of the following values of 
is there no solution 
and 
?
Solution
Notice that the two equations are lines in standard form, so the system will have no solutions if these lines never intersect. For them never to intersect, they must have the same slope and not be the exact same line. We are sure that they are not the same line, because they have different coefficients and have the same number, 
, on their right hand sides. In the case that one of the lines has an undefined slope (where it is of the form 
, where 
is some constant), then is either or 
. In either case, the other line does not have an undefined slope, so the lines eventually intersect and there must be a solution to the system of equations. Thus, we need not worry about this case. Because the lines are in standard form, we can equate the two expressions for their slope as follows:
\begin{align*}
-\frac{a-1}{a} &= - \frac{-a}{a+1} \\
-(a^2-1) &= a^2 \\
2a^2 &= 1 \\
a^2 &= \frac 1 2 \\
a &= \pm \frac 1{\sqrt2} = \pm \frac{\sqrt2}2
\end{align*}
Thus, our answer is 
.
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 40 |
Followed by Problem 42 | |
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