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
Triangular Systems:
Triangular systems of equations like the following are also eay to solve:
x = 4
x + y = 10
x + y + z = 15
x + y + z + w = 0
To solve this equation, first find x, then y, then z and then w. If rewrite this system with the equations in different order, it would still remain easy to solve:
x + y + z = 15
x + y + z + w = 0
x = 4
x + y = 10
But the order in which the equations are used to find the unknowns would change.
Instructors and Tutors: Give your students, problems with triangular systems of equations in two to several unknowns. That will build skill and confidence. It will teach your student to be opportunistic. They will have look at all the equations and all the unknowns to see which one to use or find first. The unknowns or the letters used to represent them can be permuted, so that instead of finding x first, the solution paths will find an y or w first.
Read them in order
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