Many Equations with Essentially One Variable
Example: Suppose the letters A, B, C and T satisfy the four equations
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
That means A has the same value in the first and fourth equation, and so on. Now we use the first three equations
A = 2T
B = 3T
C = 4T
to replace the A, B and C in the fourth equation
T + A + B + C = 60
That gives
T + 2T + 3T +4T = 60
In other words
10 T = 60
from which we get T = 6. The first three equations by substitution now give
A = 2T = 2*6 = 12
B = 3T = 3*6 = 18
C = 4T = 4*6 = 24
Check: we expect A =12, B = 18, C = 18 and T=6 to satisfy the original equations
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
The first three are satisfied by the way in which computed A, B and C from T while T + A + B + C = 6 + 12 + 18 + 24 = 60 by direct addition in your head, or if need-be, shame on you, with a calculator.
Remark: The fourth equation T + A + B + C = 60 is essentially an equation only in T because the other amounts A, B and C can be expressed in terms of T and thus eliminated by replacement or substitution in the fourth equation. Because the system
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
gives or reduces almost immediately a single equation in the one unknown T to solve, we say this is an essentially one variable system. We can go further. See the next example.
First Animated Example
Exercise (1) : Solve the essentially one variable system
A = 3T+ 8
B = T+ 4
C = ½T
D = 2 T - 12 and
T + A + B + 4C + D = 100
and check your answers.
The values of the unknown T may be a positive or negative proper or improper fraction, or zero. ( I do not know. I have not solved this system. I have only written it.)
Teachers: The replacement of C by ½T in the last equation forces the calculation of 4(½T) and a formal, but more likely informal, application of the associative law for arithmetic with rational or real numbers
Second Animated Example
Teachers: Substitutions or replacement in the above examples requires formal, but more likely informal, mastery of the distributive law.
Third Animated Example
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Solving Linear |
Skill in arithmetic with fractions is a must for
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