32 
BULLETIN 1351, TJ. S. DEPARTMENT OF AGRICULTURE 
Table IV c. — Solution of simultaneous equations involving {x\) price, (x 2 ) index, 
and (x 3 ) United States production plus carryover of oats l 
Equations: 
1. ax 2 2 b 12 + px 23 b 13 = px 12 
2. {px 23 b l2 ) + crx 2 3 b l3 == px 13 
Substituting actual values, as calculated in IV b. 
First equation, + 48.97 5 12 + 38.19 & 13 = + 55.27. 
Second equation, ( + 38.19 b 12 ) +300.88 b l3 = -270.95. 
612 
613 
V 
Check 
sum 
1. Bring down first equation 
+48. 97 
-1.0000 
+38. 19 « 
-. 7799 
+300. 88 
-29. 78 
+271. 10 
-1. 0000 
= +55.27 
= — 1. 12S7 
= -270.95 
= -43. 11 
= -314.06 
= +1^1585 
+142. 43' 
2. Divide by first term with sign changed, or —48.97=... 
-2. 9086 
+68. 12 
4. Multiply line (2) by +38.19= 
-111.08 
-42. 96 
6. Divide by first term with sign changed, or —271.10=.. 
+. 1585 
Change signs of values in column (p) , lines (6) and (2) , 
b 13 =- 1.1585 
b l2 = + 1.1287 + ( - 1.1585 X - 0.7799) = + 2.0322, 
Product moments : 
&i 3 Xi3= - 1.1585 X +270.95 = 313.90 
b 12 x 12 = +2.0322 X + 55.27 = 112.32 
then 
Adding, P. M. 
The coefficient of multiple correlation, R = 
4 26.22 
VP. M. 
V 
_ V 426.22 
V575.91 
= 0.86 
The estimating equation is x L = a + b 12 x 2 + b 13 x 3 
Solve for "a" by substitution as follows: +5.00 = a+ (2.0:522 X 
2.41) + (-l. 1585X3. 54). a= +4.20. 
Inserting known values for the constants: a, 6 12 , and b n , the esti- 
mating equation becomes z 1 = 4.20 + 2.03£ 2 — 1.16x 3 . 
1 The method used in solving the simultaneous equations is "The Doolittle method." See Geodesy- 
Application of Theory of Least Squares to the Adjustment of Triangulation, by O. S. Adams, 1915; also 
A. Method of Handling Multiple Correlation Problems, by H. E. Tolley and M. J. B. Ezekiel, in Journal 
American Statistical Association, December, 1923. 
