algebra

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Branch of mathematics in which the general properties of numbers are studied by using symbols, usually letters, to represent variables and unknown quantities. For example, the algebraic statement:

(x + y)² = x² + 2xy + y²

is true for all values of x and y. For instance, the substitution x = 7 and y = 3 gives:

(7 + 3)² = 7² + 2(7 × 3) + 3² = 100

An algebraic expression that has one or more variables (denoted by letters) is a polynomial equation. A polynomial equation has the form:

f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

where a_n, a_n-1, …, a₀ are all constants, n is a positive integer, and a_n ≠ 0. Examples of polynomials are:

f(x) = 3x⁴ + 2x² + 1

or

f(x) = x⁵ - 18x + 71

or

f(x) = 2x + 3

Algebra is used in many areas of mathematics – for example, arithmetic progressions, or number sequences, and Boolean algebra (the latter is used in working out the logic for computers).

In ordinary algebra the same operations are carried on as in arithmetic, but, as the symbols are capable of a more generalized and extended meaning than the figures used in arithmetic, it facilitates calculation where the numerical values are not known, or are inconveniently large or small, or where it is desirable to keep them in an analysed form.

For example, the following table shows the cost of gas for heating:

There is a connecting rule between the cost and the number of therms used. Gradient = change in cost/change in therms:

= 40 - 20/50 - 10 = 20/40 = £0.5 per therm

Cost intercept = £15 (the intercept is the standing charge).

Since this is a straight line graph, a linear equation connecting the cost and therms used can be created:

cost = 0.5 therms + 15 or c = 0.5 t + 15

A straight line graph can be represented by the general formula:

y = mx + c

where c is the y-intercept, m is the gradient, and (x,y) are the points on the line.

Order of calculation
The simplification of an algebraic equation or expression must be completed in a set order. The procedure follows the rules of BODMAS – any elements in brackets should always be calculated first, followed by power of (or index), division, multiplication, addition, and subtraction.

For example, to solve the equation:

3(2x - x - 1) = 2(x + 3 + 4)

collect the like terms and work out the brackets:

3(x - 1) = 2(x + 7)

multiply out the brackets:

3x - 3 = 2x + 14

collect the xs on the left-hand side of the equation:

3x - 3 - 2x = 14

then solve for x:

x - 3 = 14 x = 14 + 3 x = 17

Inequations or inequalities may be solved using similar rules. When multiplying or dividing by a negative value, however, the direction of the inequality must be reversed, for example: -x > 5 is equivalent to x < -5.

Quadratic equation
A quadratic equation is a polynomial equation of second degree (that is, an equation containing as its highest power the square of a variable, such as x²). The general formula of such equations is:

ax² + bx + c = 0

in which the coefficients a, b, and c are real numbers, and only the coefficient a cannot equal 0.

Some quadratic equations can be solved by factorization (see factor (algebra)), or the values of x can be found by using the formula for the general solution.

x = [-b + √(b² - 4ac)]/2a or

x = [-b - √(b² - 4ac)]/2a

Depending on the value of the discriminant b² - 4ac, a quadratic equation has two real, two equal, or two complex roots (solutions).

When b² - 4ac > 0, there are two distinct real roots.

When b² - 4ac = 0, there are two equal real roots.

When b² - 4ac < 0, there are two distinct complex roots.

Simultaneous equations
If there are two or more algebraic equations that contain two or more unknown quantities that may have a unique solution, they can be solved simultaneously as simultaneous equations. For example, in the case of two linear equations with two unknown variables, such as:

(i) 3y + x = 6 and

(ii) 3y - 2x = 6

the solution will be those unique values of x and y that are valid for both equations. Linear simultaneous equations can be solved by using algebraic manipulation to eliminate one of the variables. For example, subtracting equation (ii) from equation (i) gives:

3y - 3y + x + 2x = 6 - 6

So x = 0, and substituting this value into (ii) gives:

3y = 6

So y = 2.

Another method is to rearrange (i) to give:

x = 6 - 3y

Substituting this into (ii) gives:

3y - 2(6 - 3y) = 6

Multiplying out the brackets gives:

3y - 12 + 6y = 6

So 9y = 18, and y = 2.

‘Algebra’ was originally the name given to the study of equations. In the 9th century, the Arab mathematician Muhammad ibn-Musa al-Khwarizmi used the term al-jabr for the process of adding equal quantities to both sides of an equation. When his treatise was later translated into Latin, al-jabr became ‘algebra’ and the word was adopted as the name for the whole subject.

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