Equations in Two Variables

Definnition: An ordered pair of real numbers is a set of two elements in which one element is designated as the first element and the other the second.

Notation: (a,b) denotes an ordered pair. The first component of this ordered pair is a. The second component is b.

An equation in the two variables x and y is an equation of the form

AX + BY = C

where not both A and B are xero.

By a solution to an equation in two variables we mean an ordered pair or real numbers such that when x is replaced by the first component of the ordered pair and y is replaced by the second component a true statement results.

Example:

(2, -5) is a solution of the equation 3x - 2y = 16 since we obtain the true statement
3(2) - 2(-5) = 16 or 6 + 10 = 16 or 16 = 16 when x is replaced by 2 and y is replaced by -5.
Note (0,-8) is also a solution to the equation since 3(0) - 2(-8) = 16 is equaivalent to 0 + 16 = 16 which is equivalent to 16 = 16.

In general an equation in two varibles may have an infinite number of solutions.

Convention: In an equation in the two variables x and y, x is called the independent variable and y is called the dependent variable.

In writing a solution to an equation in x and y as an ordered pair we always list the value of the indepent variable or value of x first, then the value of the dependent variable or value of y second. If confusion arises, remember to list the values of the variables in alphabetical order since x comes before y in the alphabet.

To find solutions to an equation in two vaiables:

• Solve the equation for y. This is often referred to writing the equation in explicit form
• Substitute in a value for x and find the corresponding value for y.

Examples:

• Find the solution (3,?) to the equation 5x + y = 11.
  Write in explicit form: y = 11 - 5x.
  Substitute 3 for x: y = 11 - 5(3).
  Solve for y: y = 11 - 15 or y = -4.
  Solution: (3,-4).
  
  
• ????

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