MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - Other Middle and High School Contests
Math Prize For Girls Problems
2009 Math Prize For Girls Problems
17
17
Part of
2009 Math Prize For Girls Problems
Problems
(1)
three linear equations with a nonlinear objective
Source:
11/15/2009
Let
a
a
a
,
b
b
b
,
c
c
c
,
x
x
x
,
y
y
y
, and
z
z
z
be real numbers that satisfy the three equations \begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \\ ax + by + 42z &= 0. \end{align*}Suppose that
a
≠
13
a \ne 13
a
=
13
and
x
≠
0
x \ne 0
x
=
0
. What is the value of
13
a
−
13
+
23
b
−
23
+
42
c
−
42
?
\frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?
a
−
13
13
+
b
−
23
23
+
c
−
42
42
?
linear algebra
matrix