9
The law of inverses
The four forms of equations
Transposing
A logical sequence of statements
Transposing versus exchanging sides
The form ax = 0
Section 2:
Canceling
The unknown on both sides
Simple fractional equations
AN EQUATION is an algebraic statement in which the verb is "equals" = . An equation involves an unknown number, typically called x. Here is a simple example:
x + 4 = 10.
"Some number, plus 4, equals 10."
We say that an equation has two sides: the left side, x + 4, and the right side, 10.
Because x appears to the first power, we call that a linear equation. A linear equation is also called an equation of the first degree.
The degree of any equation is the highest exponent that appears on the unknown number. An equation of the first degree is called linear because, as we will see much later, its graph is a straight line.
The equation  that statement  will become true only when the unknown has a certain value, which we call the solution to the equation.
The solution to that equation is obviously 6:
6 + 4 = 10.
6 is the only value of x for which the statement "x + 4 = 10" will be true. We say that x = 6 satisfies the equation.
Now, algebra depends on how things look. As far as how things look, then, we will know that we have solved an equation when we have isolated x on the left.
Why the left? Because that's how we read, from left to right. "x equals . . ."
In the standard form of a linear equation  ax + b = 0  x appears on the left.
In fact, we are about to see that, for any equation that looks like this:
x + a  =  b, 
the solution will always look like this:  
x  =  b − a. 
If  
x + 4  =  10,  
then  
x  =  10 − 4  
=  6. 
The law of inverses
There are two pairs of inverse operations. Addition and subtraction, multiplication and division.
Formally, to solve an equation we must isolate the unknown, which is typically x, on one side of the equation.
ax − b + c = d.
We must get a, b, c over to the other side, so that x is alone.
The question is:
How do we shift a number from one side of an equation
to the other?
Answer:
By writing it on the other side with the inverse operation.
That is the law of inverses. It follows from the two Rules of Lesson 6.
It is in keeping with the arithmetical relationship between addition and subtraction:
10 − 6 = 4 implies 10 = 4 + 6;
and between multiplication and division:
10 2  =  5  implies  10 = 2· 5. 
And so, to solve this equation:
ax − b + c  =  d 
then since b is subtracted on the left, we will add it on the right:  
ax + c  =  d + b. 
Since c is added on the left, we will subtract it on the right:  
ax  =  d + b − c. 
And finally, since a multiplies on the left, we will divide it on the right:  
x  =  d + b − c a 
We have solved the equation.
The four forms of equations
Solving any linear equation, then, will fall into four forms, corresponding to the four operations of arithmetic. The following are the basic rules for solving any linear equation. In each case, we will shift a to the other side.
1. If x + a = b, then x = b − a.
"If a number is added on one side of an equation,
we may subtract it on the other side."
2. If x − a = b, then x = b + a.
"If a number is subtracted on one side of an equation,
we may add it on the other side."
3. If ax = b, then x =  b a  . 
"If a number multiplies one side of an equation,
we may divide it on the other side."
4. If  x a  = b, then x = ab. 
"If a number divides one side of an equation,
we may multiply it on the other side."
In every case, a was shifted to the other side by means of the inverse operation. It will be possible to solve any linear equation by applying one or more of those rules..
Transposing
When the operations are addition or subtraction (Forms 1 and 2), we call that transposing.
We may shift a term to the other side of an equation
by changing its sign.
+ a goes to the other side as − a.
− a goes to the other side as + a.
Transposing is one of the most characteristic operations of algebra, and it is thought to be the meaning of the word algebra, which is of Arabic origin. (Arabic mathematicians learned algebra in India, from where they introduced it into Europe.) Transposing is the technique of those who actually use algebra in science and mathematics  because it is skillful. And as we are about to see, it maintains the clear, logical sequence of statements. Moreover, it emphasizes that you do algebra with your eyes. When you see
x + a  =  b, 
then you immediately see that +a goes to the other side as −a:  
x  =  b − a. 
The way that is often taught these days, is to actually
write the inverse −a on both sides, draw a line, and add:
First, you will never see that in any calculus text.
And after observing the results a sufficient number of times, the student will see that the effect is to transpose +a to the other side as −a. (Lesson 6) Therefore the student should simply learn to transpose!
One may insist on the theory of balanciing the equation, but one should at least maintain the logical sequence of statements. If you say that
because we subtracted a from both sides  fine. It should not be necessary to actually write −a on both sides.
A logical sequence of statements
In an algebraic sentence, the verb is typically the equal sign = .
ax − b + c = d.
That sentence  that statement  will logically imply other statements. Let us follow the logical sequence that leads to the final statement, which is the solution.
(1)  ax − b + c  =  d  
implies  (2)  ax  =  d + b − c 
implies  (3)  x  =  d + b − c . a 
The original equation (1) is "transformed" by first transposing the terms (Lesson 1). Statement (1) implies statement (2).
That statement is then transformed by dividing by a. Statement (2) implies statement (3), which is the solution.
Thus we solve an equation by transforming it  changing its form  statement by statement, line by line according to the rules of algebra, until x finally is isolated on the left. That is how books on mathematics are written (but unfortunately not books that teach algebra!). Each line is its own readable statement that follows from the line above  with no crossings out.
In other words, What is a calculation? It is a discrete transformation of symbols. In arithmetic we transform "19 + 5" into "24". In algebra we transform "x + a = b" into "x = b − a."
Problem 1. Write the logical sequence of statements that will solve this equation for x :
abcx − d + e − f = 0
To see the answer, pass your mouse from left to right
over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
(1)  abcx − d + e − f  =  0  
implies  (2)  abcx  =  d − e + f 
implies  (3)  x  =  d − e + f . abc 
First, transpose the terms. Line (2).
It is not necessary to write the term 0 on the right.
Then divide by the coefficient of x.
Problem 2. Write the logical sequence of statements that will solve this equation for x :
(1)  2x + 5  =  27  
implies  (2)  2x  =  27 − 5 = 22 
implies  (3)  x  =  22 2 
implies  (4)  x  =  11. 
Problem 3. Solve for x : (p − q)x + r = s
Problem 4. Solve for x : ab(c + d)x − e + f = 0
x =  e − f ab(c + d) 
Problem 5. Solve for x : 2x + 1= 0
x = −½
That equation, incidentally, is in the standard form, namely ax + b = 0.
Problem 6 . Solve:  ax + b  =  0.  
x  =  −  b a 
Each of these problems illustrates doing algebra with your eyes. The student should see the solution immediately. In the example above, you should see that b will go to the other side as −b, and that a will divide.
That is skill in algebra.
Problem 7. Solve for x : ax = 0 (a0).
Now, when the product of two numbers is 0, then at least one of them must be 0. (Lesson 5.) Therefore, any equation with that form has the solution,
x = 0.
We could solve that formally, of course, by dividing by a.
Problem 8. Solve for x :
4x − 2  =  −2 
4x  =  −2 + 2 = 0 
x  =  0. 
Problem 9. Write the sequence of statements that will solve this equation:
(1)  6 − x  =  9 
(2)  −x  =  9 − 6 
(3)  −x  =  3 
(4)  x  =  −3. 
When we go from line (1) to line (2), −x remains on the left. For, the terms in line (1) are 6 and −x.
We have "solved" the equation when we have isolated x  not −x  on the left. Therefore we go from line (3) to line (4) by changing the signs on both sides. (Lesson 6.)
Alternatively, we could have eliminated −x on the left by changing all the signs immediately:
(1)  6 − x  =  9 
(2)  −6 + x  =  −9 
(3)  x  =  −9 + 6 = −3. 
Problem 10. Solve for x :  3 − x  =  −5 
x  =  8. 
Problem 11. Solve for x :
4 − (2x − 1)  =  −11. 
4 − 2x + 1  =  −11. Lesson 7 
5 − 2x  =  −11 
−2x  =  −11 − 5 
2x  =  16 Lesson 6 
x  =  8. 
Problem 12. Solve for x:
3x − 15 2x + 1  = 0. 
(Hint: Compare Lesson 5, Problem 18.)
x = 5.
Transposing versus exchanging sides
Example 1.  a + b = c − x 
We can easily solve this  in one line  simply by transposing x to the left, and what is on the left, to the right:
x = c −a − b.
Example 2.  a + b = c + x 
In this Example, +x is on the right. Since we want +x on the left, we can achieve that by exchanging sides:
c + x = a + b
Note: When we exchange sides, no signs change.
Upon transposing c, the solution easily follows:
c + x = a + b − c.
In summary, when −x is on the right, it is skillful simply to transpose it. But when +x is on the right, we may exchange the sides.
Problem 13. Solve for x :
p + q  =  r − x − s  
Transpose:  
x  =  r − s − p − q 
Problem 14. Solve for x :
p − q + r  =  s + x  
Exchange sides:  
s + x  =  p − q + r  
x  =  p − q + r − s 
Problem 15. Solve for x :
0  =  px + q  
px + q  =  0  
px  =  −  q 
x  =  −  q p 
Problem 16. Solve for x :
−2  =  −5x + 1 
5x  =  1 + 2 = 3 
x  =  3 5 
Section 2: The unknown on both sides
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Steps
Part 1 Writing Chemical Formulas of Covalent Compounds

1
Memorize the prefixes for number of atoms. In naming compounds, Greek prefixes are used to indicate the number of atoms present for each element. Covalent compounds have the first element written out completely while the second element is named with the suffix “ide”. For example diphosphorus trisulfide has a chemical formula of P2S3.[1] Below are the prefixes for 110: 1: Mono
 2: Di
 3: Tri
 4: Tetra
 5: Penta
 6: Hexa
 7: Hepta
 8: Octa
 9: Nona
 10: Deca

2
Write the chemical symbol for the first element.
 For example: Dinitrogen hexafluoride. The first element is nitrogen and the chemical symbol for nitrogen is N.
3
 For example: Dinitrogen has a the prefix “di“ which means 2; therefore, there are two atoms of nitrogen present.
 Write dinitrogen as N2.
4
 For example: Dinitrogen hexafluoride. The second element is fluorine. Simply replace the “ide” ending with the actual element name. The chemical symbol for fluorine is F.
5
2
 There are only 3 cation polyatomic ions and they are ammonium (NH4+), hydronium (H3+), and mercury(I) (Hg22+). They all have a +1 charge.
 The rest of the polyatomic ions have negative charges ranging from 1 to 4. Some common ones are carbonate (CO32), sulfate (SO42), nitrate (NO3), and chromate (CrO42).
3
 All group 1 elements at +1.
 All group 2 elements are +2.
 Transition elements will have Roman numerals to indicate their charge.
 Silver is 1+, zinc is 2+, and aluminum is 3+.
 For example: Lithium Oxide. Lithium is a group 1 element and has a +1 charge. Oxygen is a group 16 element and has a 2 charge. In order to balance the 2 charge of the oxygen, you need 2 atoms of lithium; therefore, the chemical formula of lithium oxide is Li2O.
5
 Calcium Nitride: Symbol for calcium is Ca and symbol of nitrogen is N. Ca is a group 2 element and has a charge of +2. Nitrogen is a group 15 element and has a charge of 3. To balance this, you need 3 atoms of calcium (6+) and 2 atoms of nitrogen (6): Ca3N2.
 Mercury (II) Phosphate: Symbol for Mercury is Hg and phosphate is the polyatomic ion PO4.
 For example: AgNO3 + NaCl > ?
 The cations are Ag+1 and Na+1. The anions are NO31 and Cl1.
2
 Remember to balance the charges when forming new compounds.
 For example: AgNO3 + NaCl > ?
 Ag+1 now pairs with Cl1 to form AgCl.
 Na+1 now pairs with NO31 to form NaNO3.
3
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