Many Equations with Essentially One VariableExample: Suppose the letters A, B, C and T satisfy the four equations
That means A has the same value in the first and fourth equation, and so on. Now we use the first three equations
to replace the A, B and C in the fourth equation
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 Check: we expect A =12, B = 18, C = 18 and T=6 to satisfy the original equations
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
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
and check your answers.
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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Skill in arithmetic with fractions is a must for
algebra.
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