12 02

Rules for writing equations

rules for writing equations


The law of inverses

The four forms of equations


A logical sequence of statements

Transposing versus exchanging sides

The form ax = 0

Section 2:


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 = ba.
x + 4 = 10,
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.

axb + 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?


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:

= 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

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 + ab, 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 ab, 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 axb, then x = b

"If a number multiplies one side of an equation,
we may divide it on the other side."

4. If x
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..


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 = ba.

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 = .

axb + 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) axb + c = d
implies (2) ax = d + bc
implies (3) x = d + bc .

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 = ba."

Problem 1. Write the logical sequence of statements that will solve this equation for x :

abcxd + ef = 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) abcxd + ef = 0
implies (2) abcx = de + f
implies (3) x = de + f .

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 
implies (4) x = 11.

Problem 3. Solve for x : (pq)x + r = s

Problem 4. Solve for x : ab(c + d)xe + f = 0

x = ef
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

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 (ais not equal to0).

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 = cx

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 = cab.

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 + bc.

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 = rxs
x = rspq

Problem 14. Solve for x :

pq + r = s + x
Exchange sides:
s + x = pq + r
x = pq + rs

Problem 15. Solve for x :

0 = px + q
px + q = 0
px = q
x = q

Problem 16. Solve for x :

−2 = −5x + 1
5x = 1 + 2 = 3
x = 3

Section 2: The unknown on both sides


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You mix the ingredients together, flour, butter, salt, sugar, and eggs, bake it and see that it changes into something new, cookies. In chemistry terms the equation is the recipe, the ingredients are "reactants," and the cookies are "products." All chemical equations look something like "A + B --> C (+ D..)," in which each letter variable is an element or a molecule (a collection of atoms held together by chemical bonds). The arrow represents the reaction or change taking place. To write the equations there are a number of important naming rules that you need to know.


Part 1 Writing Chemical Formulas of Covalent Compounds

  1. 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 1-10:
    • 1: Mono-
    • 2: Di-
    • 3: Tri-
    • 4: Tetra-
    • 5: Penta-
    • 6: Hexa-
    • 7: Hepta-
    • 8: Octa-
    • 9: Nona-
    • 10: Deca-
  2. 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

    Add the number of atoms as a subscript. To identify the number of atoms present for each element, you simply need to look at the prefix of the element. Memorizing the Greek prefixes will help you to be able to write chemical formulas quickly without looking anything up.[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

    Write the chemical symbol for the second element. The second element is the “last name” of the compound and will follow the first element. For covalent compounds, the element name will have a suffix of “-ide” instead of the normal ending of the element.[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

    Add the number of atoms present as a subscript.
  • For example: Lithium oxide is Li2O.
  • 2

    Recognize polyatomic ions. Sometimes the cation or anion is a polyatomic ion. These are molecules that have two or more atoms with ionic groups. There’s no good trick to remembering these, you just need to memorize them.[7]
    • 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

    Determine the valence charge of each element. The valence charge can be determined by looking at the position of the element on the periodic table. There are a few rules to keep in mind that help you identify the charges:[8]
    • 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+.
  • Once you have identified the charge of each element (or polyatomic ion), you will use these charges to determine the number of atoms present of each element. You want the charge of the compound to equal zero so you will add atoms to balance the charges.[9]
    • 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

    Practice with some examples. The best way to learn formula writing is to practice with lots of examples. Use examples in your chemistry book or look for practice sets online. Do as many as you can until you feel comfortable writing chemical formulas.
    • 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.
  • In a basic double replacement equation you will have two cations and two anions. The general equation takes the form of AB + CD --> AD + CB, where A and C are cations and B and D are anions. You also want to determine the charges of each ion.[10]
    • For example: AgNO3 + NaCl --> ?
    • The cations are Ag+1 and Na+1. The anions are NO31- and Cl1-.
  • 2

    Switch the ions to build the products. Once you have identified all of the ions and their charges, rearrange them so that the first cation is now paired with the second anion, and the second cation is now paired with the first anion. Remember the equation: AB + CD --> AD + CB.[11]
    • 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

    Write the full equation. After writing the products that will form in the equation, you can write the whole equation with both products and reactants. Keep the reactants on the left side of the equation and write the new products on the right side with a plus sign between them.
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