Expressions
such as “¾x + 2” are called linear
because the graph of the value is a straight line. This pun is based
on the derivation of the words.
This week we cover linear equations and inequalities in one variable.
Expressions
such as “¾x + 2” are called linear
because the graph of the value is a straight line. This pun is based
on the derivation of the words.
In this class, we will discuss methods of solving linear equations and inequalities, as well as how to recognize that there is no usable solution.
As our textbook points out, on page 66, linear equations can be written in the form
ax + b = 0
where a and b are constants. Linear equations can also have the forms
(ax + b) = (cx + d)
(ax + b) + (cx + d) + (ex + f) = 0
(ax + b) / e = (cx + d)
and lots of others
where all letters other than “x” are constants.
Although these equations seem different, they have one thing in common: there is no multiplication or division, except by constants. Next week, we will see the connection between this property and the linear graph.
Since the only multiplication or division is by constants, we can usually apply the strategy on page 76 of our textbook to solve the equation by isolating the variable on one side of the equation.
The exception to our textbook's strategy for solving linear equations occurs when the constant. “a” in the basic linear equation, “ax + b = 0” is zero.
If “b” is also zero, the equation reduces to 0 = 0; an identity (statement that is always true). For any other value of b, the equation “0x + b = 0” is inconsistent (that is, self contradictory or false).
Often, the original expression is so complicated, with added and subtracted terms, that it is not recognized as an identity until it is simplified. Identities are the basis for a type of magic trick; the blindfolded magician leads the audience through a series of calculations on the blackboard, and then announces the answer. Of course, as the magician knows, the solution was determined by the calculation and did not depend on the starting number.
Take a three-digit number. Multiply it by ten. Subtract the original number from the product. Now add the digits of the result of subtraction. If the sum of the digits is over ten, repeat the process of adding the digits until you are left with a single digit. That digit will be a nine.
There was some discussion in the virtual Faculty Lounge about a question raised in another math class. One of the students noticed the similarity between the format of equations and computer program statements and asked about the connection. The answer is that they are related but not identical.
The main difference is that an equation states that two expressions have the same value. The computer instruction with an equal sign tells the computer to evaluate an expression and put the value into a designated location. In Excel, the location is the cell that contains the "=". In a language like Basic, the location is named on the left side of the equal sign with a single variable. The programming languages that set values with “=” statements (also called computation statements) would not permit the statement, “3x + 5 = 2x + 3”.
The similarity of the syntax of the "=" statements in algebra and programming languages has its roots in the late 1950's with the development of Fortran (which is short for Formula Translator). Fortran was developed to automatically generate number-processing instructions that a computer could understand. Before Fortran, programmers had to translate expressions such as “12(2 x + 4)” manually into a series of adds, multiplies etc. in the correct order (in this example 2 times x plus 4, then multiplied by 12. Since the language was designed for mathematicians, the grammar of computation statement was copied from algebra. IBM even added hole-combinations for parenthesis to their punch card equipment to accommodate the new language.
Most programming languages developed since the release of Fortran copied the same format for their computational statements. When spreadsheets were developed, the same formats were still available (this design gave the first users one less thing to learn).
Just
as equations state that two quantities are equal, inequalities state
that one is greater (or less) than the other.
There are two ways of visualizing an
inequality such as “(x/2 + 1) ≥
2”. One way graphs the line y=(x/2 + 1), takes the portion for
which y is greater than or equal to 2, and projects this portion onto
the number line. The other way is to solve the inequality by
isolating the x onto one side
(x/2 + 1) ≥ 2
(x/2) ≥ 1 subtract 1 from both sides
x ≥ 2 multiply both sides by 2
The first method helps in visualizing the problem; the second achieves the result more quickly.
Page 102 of our textbook has an overall strategy for solving word problems. The most important step is understanding the conditions described in the problem and what information is requested; you don't want to calculate the speed of an automobile if the question was “What does the car weigh”?
Very often, the type of problem is familiar and you know that the mathematic formula has been discussed, but you are not sure where. The index in the back of the textbook is rather helpful in such situations; our textbook also has a directory of applications in the front cover. When you find the applicable formula, make sure that the numbers in the original problem are inserted in the correct locations. If you interchange weight and height while calculating the body mass index, the result could frighten you into an unnecessary crash diet.
Another hazard in applying formulas is in unit of measure. A recent Mars probe crashed because of confusion over whether the altitude was in feet or in meters. Wherever feasible, keep the unit of measure with the number. If the result of calculating the speed is in hours per mile, you can adjust accordingly.
Example 3 on Page 81 of our textbook suggests multiplying both sides of the equation by the lowest common denominator (LCD); what is the drawback of using another common denominator (not the lowest)? Can you think of a problem for which a larger denominator would make sense?
Construct the equation for problem 73 on page 117. How would you solve it?
Come up with a problem from your daily life that can be solved using a linear inequality. Show how you solve the inequality and explain what the solution space is.*
Remember
the Pythagorean theorem? “In a right triangle, the
square of the length of the hypotenuse is equal to the sum of the
squares of the lengths of the other two sides.” We can
simplify this statement by writing the equation: c² = a²
+ b²,
where c is the hypotenuse (long side) and a and b
are the other two sides.
Invent a complex word problem
and translate it into a mathematical equation. Explain how the
words in your problem map into mathematical operations.*
* Suggested by Dr. Cook.