Linear equations
      Linear equation
      Solving linear equations
      Applications of linear equations, word problems
         Numbers word problem
         Geometry and physics word problems
Linear equation
An equation is linear if it can be written in the form 
ax = b,    where x is the variable and a and b are constants.
A linear, means the variable x appears only to the first power.
The solution of an equation is the value of the variable x that makes the equation a true statement.
Solving linear equations
We solve an equation by transforming it through a sequence of equivalent equations, statement by statement, until x finally is isolated on the left.
Two equations that have the same solutions are called equivalent.
Shift a number from one side of an equation to the other, by writing it on the other side with the inverse operation.
When the operations are addition or subtraction a term may be shifted to the other side of an equation by changing its sign.
After solving an equation check the solution by substituting them in the original equation. 
Examples:   a)   -1 - (2 - x) = 2  - (3x + 1)
                           -1 - 2 + x = 2  - 3x - 1
                                x + 3x = 2 + 2
                                      4x = 4 | 4
                                        x = 1
Applications of the linear equations, word problems
Word problems - translate the words into mathematics 
Numbers word problem
Example:  The sum of two digits of a two-digit number is 10. If the digits are reversed, the new number is 
                  four less then five times the original number. Find two-digit number.
Solution:   Let x be tens place value digit then, 10 - x is units digit, thus given number is  10x + (10 - x).
By reversing digits we get 10 (10 - x) + x.
Hence given problem yields:    10 (10 - x) + x = 5 [10x + (10 - x)] - 4
                                                100 - 10x + x = 5 [9x + 10] - 4
                                                        100 - 9x = 45x + 50 - 4
                                                               54x = 54 | 54
                                                                   x = 1       thus, the two-digit number is:  10x + (10 - x) =19.
Geometry and physics word problems
Example:  A rectangle has the perimeter 90 cm and the width 13 cm less then the length. 
                 What is the area of the rectangle?
Solution:
 
  a = b + 13 = 16 + 13 = 29 cm
A  = a b = 29 16 = 464 cm2   
the perimeter P of a rectangle:
P = 2 (a + b) = 90
      2 (b + 13 + b) = 90 | 2
           2b + 13 = 45
                   2b = 32
                         b = 16 cm
Example:  After a car A started traveling at a speed of 90 km/h, half-hour later from the same place in the 
                 same direction starts car B traveling at 120 km/h.  When will car B reach the car A? 
  Solution:  The cars are traveling the same distance  d = r t .   distance = rate x time
                  The car A takes x hours, and the car B,  x - 1/2 hours.
                  Thus,     90 x = 120 (x - 1/2) | 30
                                    3x = 4x - 2
                                      x = 2 hours,    the car B will reach the car A after  x - 1/2 = 2 - 1/2 = 3/2 hours.
Beginning Algebra Contents C
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