# linear equations and word problems

Please see the attached file for the fully formatted problems.

1. Translate the sentence to an equation and solve it.

Six less than eight times a number is equal to that number added to one.

2. The length of a rectangular mailing label is 3 centimeters less than twice the width. The perimeter is 54 centimeters. Find the dimensions of the label.

3. The perimeter of a rectangular park is 22,735.8 inches. The length of the park is 9,890.1 inches longer than the width. What is the length of the park?

4. A student needs 10% hydrochloric acid for a chemistry experiment. How much 2% acid should be mixed with 200 ml of 30% acid to get a 10% solution?

5. A student needs 20% hydrochloric acid for a chemistry experiment. How much 3% acid should be mixed with 200 ml of 30% acid to get a 20% solution?

6. How many gallons of 90% antifreeze solution must be mixed with 80% of 20% antifreeze to get a mixture tht is 80% antifreeze?

7. Students stuff envelopes for extra money. Their initial cost to obatin the confirmation for the job was $600. Each envelope costs $0.02 and they get paid $0.08 per envelope stuffed. Let x represent the number of envelopes stuffed.

a) Express the cost C as a function of x.

b) Express the revenue R as a function of x.

C) Determine analytically the value of x for which revenue equals cost.

8. Find the constant of variation k and the undertermined value in the table if y is directly proportional to x.

9. The pressure exerted by a certain liquid at a given point varies directly as the depth of the point beneath the surface of te liquid. The pressureat 80 feet is 560 pounds per square inch. What is the pressure at 30 feet?

10. A person 4-ft tall casts a shadow 3-ft long. At the same time, a nearby tree casts a shadow 22-ft long. Find the height of the tree.

11. Hooke's law. The distance d when a springis stretched by a hanging object varies directly as the weight w of the object. If the distance is 61 cm when the weight is 3 kg. What is the distance when the weight is 9 kg?

12. To determine the number of trout in a lake, a conservationist catches 115 trout, tags them and throws them back into the lake. Later, 26 trout are caught, 13 of them are tagged. How many trout are in the lake?

13. A company finds it can produce 25 heaters for $6100, while producing 30 heaters costs $7200. Express the cost, y, as a linear function of the number of heaters, x. Determine the cost to produce 40 heaters.

14. Cost for copies is a linear function of the number of copies. If 200 copies cost $43.00, and 125 copies cost $30.25, write a formula for copy cost a linear equation of the number of the copies. Then find how much it would cost to make 200 copies.

15. A 20 year old wonan whose resting heart rate is 65 beats per minute should not exceed a maximum heart rate of 173. A 20 year old wonan whose resting heart rate is 90 beats per minute should not exceed a maximum heart rate of 178.

a) List the data points (r, M)

b) Find the linear equation which fits the data.

c) Use the equation to predict the maximum heart rate for a woman whose resting heart rate is 70 beats per minute.

16. In the 1990s, a company had a profit per share given by the equation P=0.30x + 4.50, where x ranges from 0 to 9 corresponding to the years 1990 to 1999. What was the profit per share in 1996?

17. Solve for g. q = 3g + 3w

18. Solve the formula for indicated variable. P = a + 4b + 4c, for a

19. Solve for a. D = 1/5 * h(a-b)

20. Solve for u. k = u + unk

21. Bill invests in a plan that has an APR of 3%. He invests four times as much in a plan that has an APR of 8%. If the total interest from the investments is $805 after one year, how much was invested in each plan?

#### Solution Preview

Please see the attached file for the complete solution.

Six less than eight times a number (denoted as x) is 8x - 6

So the equation is

8x - 6 = x + 1

Move x terms to the left-hand side and constant to the right hand side:

8x - x = 1 + 6

7x = 7

Divide 7 from both sides:

x = 1

Answer: D, solution is 1.

Let x be the width, then the length is 2x - 3.

The perimeter is 2*width + 2 * length = 2x + 2(2x - 3) = 2x + 4x - 6 = 6x - 6

6x - 6 = 54

6x = 54 + 6

6x = 60

x = 10

so the width is 10 cm.

The length = 2 * 10 - 3 = 17 cm.

Let x be width, the length is 9890.1 + x

The perimeter is then 2x + 2(9890.1 + x) = 2x + 19780.2 + 2x = 4x + 19780.2

4x + 19780.2 = 22735.8

4x = 22735.8 - 19780.2

4x = 2955.6

x = 738.9

so the length is 9890.1 + 738.9 = 10629 inches.

Let x be the volume of 2% acid needed to be added to the 30% acid.

The HCl in x ml 2% is x * 2% = 0.02x ml

The HCl in 200 ml 30% acid is 200 * 30% = 60 ml

After the mixing, HCl is 0.02x + 60 ml.

The total volume of the solution is x + 200 ml.

For this (x + 200) ml of 10% ...

#### Solution Summary

The solution shows how to set up and solve the linear equations for word problems. The solution is detailed and well presented