Open Sentences In Two Variables

Objective: To find solutions of open sentences in two variables and to solve problems

involving open sentences in two variables

A solution of an open sentence with variables x and y is an ordered pair

• (2, -1) is a solution to the equation 3x + 2y = 4 because, 3(2) + 2(-1) = 4

• The set of all solutions to the open sentence is called the solution set.

Solve the equation y =4x – 6if the domain of x is {-2, -1, 0, 1}

x 4x – 6 y solution

-2 4(-2) – 6 -14 (-2,-14)

-1 4(-1) – 6 -10 (-1,-10)

0 4(0) – 6 -6 (0,-6)

1 4(1) – 6 -2 (1,-2)

You have one dollar in quarters and dimes. Find all the possible coin combinations.

• The problem asks for the number of quarters and dimes that make up one dollar.

• Let q equal the number of quarters and d equal the number of dimes

• The total value of the quarters and dimes will then equal: .25q + .10d

.25q + .10d = 1.00

• Solve for one variable

• -.25q + .25q + .10d = -.25q + 1.00.10d = -.25q + 1.00.10d/.10 = -.25q/.10 + 1.00/.10d = -2.5q + 10

• Keep two things in mind: both d and q must be whole numbers, q < 4 and d < 10

Solutionq -2.5q + 10 d (q,d)

0 -2.5(0) + 10 10 (0,10)

1 -2.5(1) + 10 7.5 ø

2 -2.5(2) + 10 5 (2,5)

3 -2.5(3) + 10 2.5 ø

4 -2.5(4) + 10 0 (4, 0)

The solution set must then be:

• {(0,10), (2, 5), (4, 0)}