Equations are mathematical statements that assert the equivalence of
two expressions. An equation is represented by any expression
containing a relative operator (Figure 3.6), most often the equality operator.
More precisely, Myron uses the term equation to mean an expression
whose dominant relative operator is the equality operator.
The term relative equation is used to mean an expression whose dominant relative
operator is other than the equality operator. The dominant relative operator is
the rightmost unparenthesized relative operator.
The non-dominant relative operators will yield Boolean results when evaluated.
Figure 3.6 Relative Operators
Subexpressions can be moved from one side of the equation to the
other, but equality (or relativity) is always preserved. Figure 3.7 contains
some examples.
Equation
Action
Result
(2⋅x+3)⋅x+.{4}=10
→
(2⋅x+3)⋅x=-4+10
(2⋅x+3)⋅x+4=.{10}
←
(2⋅x+3)⋅x+4-10=0
(2⋅x+3)⋅.{x}=-4+10
→
2⋅x+3=(-4+10)÷x
2⋅x+3=.{(-4+10)÷x}
←
2⋅x+3-(-4+10)÷x=0
Figure 3.7 Commute across equation
3.5.1.1 Solving a simple linear equation
A
linear equation
in one variable consists of an asserted equivalence between two
expressions with one unknown. Finding a solution involves isolating
the terms containing the unknown on one side of the equivalence
operator, with all other terms on the other side. For example, the
linear equation
x-4=10
can be solved by selecting 4, moving it to the right, and simplifying.
Try to solve the equation
5⋅x-8=3⋅x+6.
3.5.1.2 Solving a system of linear equations
Given
two equations in two unknowns
, it is possible to isolate one unknown on each side of the
equivalence operator. For example, consider the equations (1)
2⋅x=y-3
and (2)
-3⋅x=y-2. The
2
in equation (2) can be moved left, isolating y:
-3⋅x+2=y. Then
y
is selected in the equation (1) and Substitute provides
-3*x+2
as a candidate. After substitution,
x
is isolated in the equation (1) to yield
x=-1÷5. Then
x
is selected in the equation (2) and Substitute provides
-1÷5
as a candidate. After Simplify ,
y=13÷5.
These techniques show how to solve equations using standard algebraic manipulation.
A simpler method involves Select , Gather and Solve .
