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Solving Linear |
Read them in order
2.1 Use of the Equal Sign
Here are a few words about the equal sign. The equal sign = is used to say or suggest the following.- two different symbols (or expressions) are shorthand for the same number and quantity.
- two different calculations or expressions give the same result when done, or
- the value of a number or quantity can be computed using another expression.
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Here the first equation or equality holds (meaning is true) since both 4+5 and 7+2 are expressions giving the value 9. The second equation r2 = r·r always holds, no matter what value you give to r. It tells us how to compute the number or quantity described by the expression r2. The third equation 3x+1 = x+7 holds (is true) when and only when x = 2. When x has a value other than 2, the statement (suggestion or assertion) that 3x+1 gives the same result as x+7 is false. The fourth statement x+4 = x+6 is always false. No value given to (or substituted for) x will make this statement true. Adding 4 and adding 6 to the same number give different results, no matter what the number is.
Abuse of Equal Sign
The solution of the equation
| 3 4 |
= | x 4 |
is given x =3. But it is an error, a mistake, a major misuse of the equal sign, to insert an = 3 besides the x in the above equation and thus write
| 3 4 |
= | x =3 4 |
in place of writing x = 3. While a person who writes
x = 3
3
may mean x = 3, for mathematicians
x = 3
3
actually means a third of x is 3.
Proper Format: For the sake of observable skill development, the statement of the equation to solve
| 3 4 |
= | x 4 |
should have been followed by a steps like the following
| 4 × | 3 4 |
= 4 × | x 4 |
and then the conclusion
3 = x,
and then the conclusion
x = 3.
The difference between the last two equations is cosmetic: We prefer to say x is 3 or equals 3 in place of saying 3 is or equals x. In general, when two different expressions are opposite sides of an equal sign,
first expression = second expression
then should be read as follows. The first expression in the circumstances at hand has or will have when evaluated, the same value as the second expression, when or if it too is evaluated.
Format for Evaluation of algebraic and arithmetic expressions, a standard.
Note vertical alignment of equal signs
| Steps: |
A Simple Example |
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In all the foregoing, keep the = signs vertically aligned, At this stage, students may not know how to use the equal sign.=, or see the necessity for it, but the format above introduces a proper use of the equal sign. Mastering this format ensures proper use of the equal sign is ensured. Teaching this format show students how to present their work or reasoning on paper in a systematic instead of ad hoc manner.. Note: Students may do work in their head. But teachers cannot read students mind. The use of this format provide an observable skill that can be seen and verified, or corrected. Teaching may provide students with food for thought, but in the end in practice, skill and knowledge development has to be observable to be believed and guided. |
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Another Formatting Example
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Observe how curve arrow shows the next step in the
calculation. |
Formatting Advantages: The above format for formula usage or
evaluation provides a model for students to follow not for rectangle
area evaluation, and also for the evaluation of formulas triangle,
trapezoidal, parallelogram and circle area and perimeter. There-in
lies a model for showing work and for showing and recording
comprehension in mathematics, science and further quantitative arts and
disciplines, where formula evaluation questions.