Linear Equations and Word Problems Solving linear equations
Linear equation in one variable
An equation in x is a statement that two algebraic expressions are equal.

An equation that is true for all values of its variables is called an identity (identical equation).

For example,       (a - b)(a + b) = a2 - b2.
An equation that is true for the certain value of the variable (the root of the equation) is called a conditional equation.
A linear equation is any equation that can be written in the form
ax + b = 0,  where a and b are constants (fixed real numbers) with a not 0 and x is a variable.
A linear, means the variable x appears only to the first power.
The variable, as unknown quantity of which the values are to be found, may be denoted other than x.
To solve an linear equation in x means to find the value of x for which the equation is true. Such value is a solution.
For example, x = 2 is a solution of the equation 3x - 6 = 0 because 3 · 2 - 6 = 0 is a true statement.
Solving linear equations
A linear equation has exactly one solution.
Simplify both sides of the equation by removing symbols of grouping (like brackets or fractions), combining like terms, or simplifying fractions on one or both sides of the equation.
To solve an equation, isolate its variable on the left side of the equation by a sequence of equivalent simpler equations, each of which have the same solution as the original (given) equation.
To get all terms with variable in them on one side of the equation and all constants on the other side use the following methods:
- add or subtract the same quantity to or from each side of the equation,
- when the operations are addition or subtraction a term may be shifted to the other side of an equation by changing its sign,
- multiply or divide each side of the equation by the same nonzero quantity,
- divide (or multiply) both sides of the equation by the coefficient of the variable to make its value unity.
To solve an equation involving fractional expressions, find the least common denominator of all terms and multiply every term of the equation by the common denominator.
When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous solution is one that does not satisfy the original equation.
If there are variables in the denominators of the fractions identify values of the variable which will give division by zero as we have to avoid these values in the solution.
After solving an equation, verify the answer by plugging the result into the original equation.
Examples:   a)   23 - [9x - (17 + 5x) - 6] = 10x - [11  - (2x - 7) + 21x] + 9
23 - [9x - 17 - 5x - 6] = 10x - [11  - 2x + 7 + 21x] + 9

23 - 4x + 23 = 10x - 19x - 18 + 9

5x = - 55 | ¸5

x = - 11

b)   3(x - 1) · (x + 5) + x · (x - 4) = 4x(3 + x) + 5

3(x2 + 4x - 5) + x2 - 4x = 12x + 4x2 + 5
3x2 + 12x - 15 + x2 - 4x = 12x + 4x2 + 5
- 4x = 20 | ¸(- 4)
x = - 5
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