Equations

An equation is a statement of equality. The statement may be true for¹ all values of the letters.

The value of the letters for which the equation is true is the root or solution of the equation.

When a statement of equality of this kind is given, our interest is in finding² the value of the letter for which it is true. The following rules aid in finding the root.

1. The roots of an equation remain the same if the same expression is added to or subtracted from both sides of the equation.

2. The roots of an equation remain the same if both sides of the equation are multiplied or divided by the same expression other than zero and not involving the letter whose value is in question³.

The equation 2x = 4 where x is the unknown, is true for x=2. To illustrate the first of the above two rules, add 5x to both sides of the equation 2x = 4. We get 2x+5x = 4+5x which, like equation 2x = 4 is true for only x = 2. To illustrate the importance of the restriction in the second of the above two laws, multiply both sides of the equation by x and get (2x)x = (4)x which is true not only for

x = 2 but also for x = 0.

It is always a good plan to check the accuracy⁴ of one's work by substituting the result in the original equation to see whether the equation is true for this value.

Rule 1 is applied very frequently. It is, therefore, desirable to state it in a way which mechanizes its application.

If the equation 4x = 28-3x is given, in applying Rule 1, 3x may be added to both sides of the equation, yielding 4x+3x = 28-3x+3x = 28.

The result of the operation consists in omitting⁵ the term +3x to the left side. We call this operation transposition of the term 3x. This operation is an application of Rule 1 and may be explained in the following way:

Any term of one side of an equation may be transposed to the other side if its sign is changed.

Example. Find the value of x which satisfies 3x+ 7(4-x)+6x = 15. Clearing of parentheses and combining terms:

3x + 28 – 7x+ 6x = 15,

2x+28 =15.

Transposing +28 from the left side:

2x = 15 - 28,

2x = - 13.

Dividing each side by 2, according to Rule 2:

2x/2=-13/2; x=-13/2.

An equation which can be reduced to the form ax + b =0 (a≠0), is called a linear equation in x.

To solve an equation containing fractions, first reduce each fraction to its lowest terms. Then multiply each side of the equation by the least common denominator of all the denominators. This process is called clearing of fractions.

A quadratic equation is one which can be reduced to the form ax²+bx+c = 0 (a≠0) where a, b and c are known and x is unknown.

Notes:

¹ may be true for – может быть действительным для, пригодным для, верным, справедливым

² our interest is in finding – нам интересно найти

³ in question – искомое (которое неизвестно)

⁴ to check the accuracy – чтобы проверить точность

⁵ consists in omitting – состоит в устранении