Answer:
see attached
Explanation:
Given the pictorial equations, you want to know the single-digit values of each of the images.
Equations
There are 6 equations in 5 unknowns, but no numerical values are assigned. Solving the equations will only give ratios of values. From those, we need to determine the numerical values that will keep all numbers in the range 1–9.
Let the icons be represented by letter variables as follows:
- hat = h
- boot = b
- sock = s
- tennis shoe = t
- mitten = m
Rewriting the equations to standard form, we have ...
h +b -m -s = 0 . . . . . [eq1]
h -b +t = 0 . . . . . . . . [eq2]
b -3s = 0 . . . . . . . . . [eq3]
h -s -t +m = 0 . . . . . [eq4]
h +2s -m = 0 . . . . . . [eq5]
h +b +s -2t = 0 . . . . . [eq6]
Solution
We can add [eq4] to [eq1] and [eq5] to eliminate the m variable.
(h +b -m -s) +(h -s -t +m) = 0 ⇒ 2h +b -2s -t = 0 . . . . [eq7]
(h +2s -m) +(h -s -t +m) = 0 ⇒ 2h +s -t = 0 . . . . . . . . . [eq8]
Adding [eq2] to [eq6], and [eq7] will eliminate the b variable.
(h +b +s -2t) +(h -b +t) = 0 ⇒ 2h +s -t = 0 . . . . same as [eq8]; no help
(2h +b -2s -t) +(h -b +t) = 0 ⇒ 3h -2s = 0 . . . . . [eq9]
Now, we have 3 equations in 4 unknowns:
b -3s = 0 . . . . . . . [eq3]
3h -2s = 0 . . . . . . [eq9]
2h +s -t = 0 . . . . . [eq8]
The smallest positive integer values that will satisfy [eq9] are ...
h = 2, s = 3
Using these in the remaining equations gives ...
b -3·3 = 0 ⇒ b = 9
2·2 +3 -t = 0 ⇒ t = 7
And we can find m using [eq5]:
m = h +2s = 2 +2·3 = 8
Then the solution is ...
- hat = 2
- boot = 9
- sock = 3
- tennis shoe = 7
- mitten = 8