Solutions for Session 6, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6| A7

Problem A1

 a. This equation is always true. b. This equation is always false. c. This equation is true only if y = 8. It is false if y is any other number. d. This equation is true for an infinite number of pairs of values for x and y. For example, x = 1 and y = 4 make the equation true, while x = 6 and y = 3 make it false. e. This equation is always true. f. This equation is always false.

Problem A2

 a. There are no variables in this equation, so there is no variable to solve for. This equation is always true. b. There are no variables in this equation, so there is no variable to solve for. This equation is always false. c. The solution set is {8}, because y = 8 is the only number that would make the equation true. d. There are an infinite number of pairs of solutions: x = 1, y = 4 is one. The solution set is {(1, 4), (2, 5), (3, 6), (4, 7), Š}, although there are many other solutions which are not integers; (1 1/2, 4 1/2) is one. e. There are an infinite number of solutions -- x can be any number, and the equation will always be true. The solution set is {Š, -3, -2, -1, 0, 1, 2, 3, Š}. As in the previous question, there are many more solutions that are not integers. f. There are no solutions to this equation, because the right side is always 1 larger. The equation is always false.

Problem A3

 a. One answer is to replace the "?" with 31. Another is to replace it with 21 + 10 or any quantity that must equal 31. b. Replace the "?" with 32, or any quantity not equal to 31. c. Replace the "?" with a variable letter, like x.

 Problem A4 The solution is that 1 rectangle will balance with 1 square. While it would also be correct to say that 1 rectangle will balance with 1/3 of a circle, this answer does not reflect use of the second scale.

 Problem A5 One possible answer is that, for scale D, the circle is equivalent to 5 cubes. The cone is equivalent to 4 cubes, and the cylinder is equivalent to 6 cubes.

 Problem A6 One possible answer is that, for scale D, the cylinder and circle are equivalent to 1 cube. The cylinder is equivalent in weight to 3 circles, and the cube is equivalent to 4 circles.

 Problem A7 Each scale is balanced only if the quantities on each side of the scale are equivalent. In the same way, an equation is true only if the quantities on each side of the equal sign are equivalent. When variables are used, the scale may be balanced always (as it is in equation (e)), sometimes (as in equations (c) and (d)), or never (as in equation (f)).

 Session 6: Index | Notes | Solutions | Video