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Part Two 311
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12
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Question 12-1
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Suppose a is a constant and x is a variable for which we want to solve We see the following equation: x 2a = 5a How can we get an equation with x alone on the left side of the equals sign and a multiple of a alone on the right
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Answer 12-1
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We can morph the equation as follows: x 2a = 5a (x 2a) + 2a = 5a + 2a x = 7a
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Question 12-2
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Suppose b is a constant and z is a variable for which we want to solve We see: z + 3b = b/2 How can we get an equation with z alone on the left side of the equals sign and a multiple of b alone on the right
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Answer 12-2
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We can morph the equation as follows: z + 3b = b/2 2(z + 3b) = 2(b/2) 2z + 6b = b (2z + 6b) 6b = b 6b 2z = 5b z = ( 5/2)b
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Question 12-3
When we divide both sides of a first-degree equation by any expression containing the variable, there s a hidden risk What is that risk
Answer 12-3
The expression containing the variable must not be equal to 0, or we ll run into trouble of some sort The fact that the expression contains an unknown means that we can t be sure it s nonzero until we know the solution to the equation! When trying to solve a first-degree equation, therefore, it s best to avoid dividing through by any expression that contains the variable
312 Review Questions and Answers Question 12-4
Suppose we see the following equation where a, b, c, and d are constants and x is the variable: 3abx = 7x /(c + d ) To be sure this equation makes sense, we must restrict the values of the constants How
Answer 12-4
We must require that c d That is, we can t let c and d be additive inverses If c and d happen to be additive inverses, then we will find ourselves dividing by 0 on the right side of the equation
Question 12-5
Which of the following equations are first-degree equations in one variable Which are not Letters a, b, and c represent constants Letters x, y, and z represent variables x + 3a + 2b = c (c + a)x = b (x + y)a = b y2 + ay + b = c x+y+z=0
Answer 12-5
The first and second equations are first-degree equations in one variable The third and fifth equations aren t, because they contain more than one variable The fourth equation isn t, because it contains the square of the variable
Question 12-6
When using letters to represent constants, or when reading texts in which letters are used to represent constants, we have to be clear and careful with the context Why
Answer 12-6
Some letters are widely used to represent specific physical, chemical, or mathematical constants In physics, c represents the speed of light in a vacuum; N is often used to represent a chemical constant called the Avogadro constant; e is commonly used to represent the exponential constant We don t want to let c be a general constant if we re writing about relativity theory, or N stand for a general constant if the subject happens to be chemistry, or e indicate a general constant if the discussion involves exponential functions
Part Two 313 Question 12-7
Suppose a, b, and c are constants, and x is a variable How can we manipulate the following equation so it contains x all by itself on the left side, and an expression containing the constants without x on the right side, showing and justifying every step in an S/R table x 2c + 7 = 4x + 2ax + b + c
Answer 12-7
We can morph the equation as described in Table 20-1 The end result is x = (b + 3c 7) / ( 3 2a)
Question 12-8
We had better be careful about something in the last step of the process shown by Table 20-1 What s that
Answer 12-8
We must require that 2a 3 If we do not impose that restriction, we allow for the possibility of division by 0 in the last step
Question 12-9
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