1. Bruce, an appliance salesman, earns a commission of $50 for each washing machine he sells and $100 for each refrigerator. Last month he earned $500 in commissions. Find all possibilities for the number of each kind of appliance he could have sold.

2. Luis has 95 cents in dimes and quarters. Find all possibilities for the number of each type of coin he could have.

3. A certain quadrilateral has three sides of equal length and its perimeter is 19cm. Find all integral possibilities for the lengths of the sides in centimeters.

4. An isosceles triangle has perimeter 15m. Find all integral possibilities for the lengths of the sides in meters.

In the following problem, (a) choose two variables to represent the numbers asked for, (b) write an open sentence relating the variables, and (c) solve the open sentence and give the answer to the problem. (Include solutions in which one of the variables is zero.)

Luis has 95 cents in dimes = D SAY and quarters = Q SAY. Find all possibilities for the number of each type of coin he could have.
TOTAL NUMBER OF DIMES AND QUARTERS = D+Q
VALUE OF DIMES = D*10=10D CENTS
VALUE OF QUARTERS =Q*25 = 25Q CENTS
TOTAL VALUE = 95 CENTS = 10D+25Q CENTS...
TO SOLVE WE SHOULD HAVE ANOTHER PARAMETER
LET THE TOTAL NUMBER OF COINS = 5
THEN D+Q = 5

D=5-Q
10(5-Q)+25Q=95
50-10Q+25Q=95
15Q=95-50=45
Q=3
D=2 IS ONE SOLUTION SET
IF YOU WANT ALL POSSIBILITIES THEN
Q=1....D=7
Q=3....D=2
THERE IS NO SOLUTION FOR D OR Q = 0 AS PER YOUR GIVEN DATA

1. To find the possibilities for the number of each kind of appliance Bruce sold, we can set up a system of equations. Let's say he sold x washing machines and y refrigerators. We know that he earned $50 for each washing machine sold, so the total earnings from washing machines is 50x. Similarly, the total earnings from refrigerators is 100y. Since his total commissions were $500, we can set up the equation:

50x + 100y = 500

To find the possibilities for x and y, we need to find integer solutions for this equation.

We can start by dividing the equation by 50 to simplify:

x + 2y = 10

Now, we can go through different possible values for y and find the corresponding values for x.

- If y = 0, then x = 10.
- If y = 1, then x = 8.
- If y = 2, then x = 6.
- If y = 3, then x = 4.
- If y = 4, then x = 2.

So, the possibilities for the number of washing machines and refrigerators Bruce could have sold are:
- 10 washing machines and 0 refrigerators.
- 8 washing machines and 1 refrigerator.
- 6 washing machines and 2 refrigerators.
- 4 washing machines and 3 refrigerators.
- 2 washing machines and 4 refrigerators.

2. To find the possibilities for the number of dimes and quarters Luis could have, we can set up a similar approach. Let's say he has x dimes and y quarters.

For dimes, each one is worth 0.10 dollars or 10 cents. So, the total value of dimes is 0.10x. For quarters, each one is worth 0.25 dollars or 25 cents. The total value of quarters is 0.25y. Since Luis has a total of 95 cents, we can set up the equation:

0.10x + 0.25y = 0.95 (in dollars)

To avoid decimal numbers, we can multiply the entire equation by 100:

10x + 25y = 95

Now, let's find the possibilities for x and y.

- If y = 0, then x = 9.
- If y = 1, then x = 7.
- If y = 2, then x = 5.
- If y = 3, then x = 3.
- If y = 4, then x = 1.

So, the possibilities for the number of dimes and quarters Luis could have are:
- 9 dimes and 0 quarters.
- 7 dimes and 1 quarter.
- 5 dimes and 2 quarters.
- 3 dimes and 3 quarters.
- 1 dime and 4 quarters.

3. To find the integral possibilities for the lengths of the sides of the quadrilateral, we know that it has three sides of equal length and a perimeter of 19 cm.

Let's denote the length of each side as x. Since the quadrilateral has three sides of equal length, we can set up the equation:

3x = 19

Simplifying the equation, we find:

x = 19 / 3 = 6.33 (approximately)

Since we are looking for integral possibilities, we can round down to the nearest whole number:

x = 6

Therefore, the only integral possibility for the length of the sides is 6 cm.

4. Similarly, for the isosceles triangle, let's denote the length of the equal sides as x. The perimeter of the triangle is 15 m.

The perimeter of an isosceles triangle is given by:

2x + base = perimeter

Since the triangle has two equal sides, we can rewrite the equation as:

2x + x = 15

Simplifying the equation, we have:

3x = 15

Dividing both sides by 3, we find:

x = 5

Therefore, the only integral possibility for the length of the equal sides is 5 m.