Difference between revisions of "1977 AHSME Problems/Problem 21"
(Created page with "== Problem 21 == For how many values of the coefficient a do the equations <cmath> |
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\textbf{(C)}\ 2 \qquad | \textbf{(C)}\ 2 \qquad | ||
\textbf{(D)}\ 3 \qquad | \textbf{(D)}\ 3 \qquad | ||
− | \textbf{(E)}\ \infty</math> | + | \textbf{(E)}\ \infty</math> |
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==Solution== | ==Solution== | ||
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+ | Subtracting the equations, we get <math>ax+x+1+a=0</math>, or <math>(x+1)(a+1)=0</math>, so <math>x=-1</math> or <math>a=-1</math>. If <math>x=-1</math>, then <math>a=2</math>, which satisfies the condition. If <math>a=-1</math>, then <math>x</math> is nonreal. This means that <math>a=-1</math> is the only number that works, so our answer is <math>(B)</math>. | ||
− | + | ~alexanderruan |
Revision as of 23:22, 1 January 2024
Problem 21
For how many values of the coefficient a do the equations have a common real solution?
Solution
Subtracting the equations, we get , or , so or . If , then , which satisfies the condition. If , then is nonreal. This means that is the only number that works, so our answer is .
~alexanderruan