Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Algebra - Equations - Simultaneous equations - Solve by graphing.
Test Yourself 1.


 

Use the following graphs to solve the given pair of simultaneous equations:

1. The equations are:

y = 2x + 4

y = 1 - x

Answer.The POI is (-1, 2).

2. The equations are:

y = 2 - x

y = 2x - 1

Answer.The POI is (1, 2).
3.

Answer.The POI is (1, -4).

 4.

Answer.The POI is (1, -3).

 

Graph each of the following pairs of equations to find the solution for their point of intersection:

1. y = 2x + 7

y = 4x + 13

Answer.The POI is (-3, 1).

 2. b = 2a

a + b = 3

Answer.The POI is (1, 2).
3. 2x + y = -2

4x - y = 8

Answer.The POI is (1, -4).

 4. 2p - q = 5

5p - 2q = 8

Answer.The POI is (1, -3).
5. 2n = 3m + 6

4m - n = 2

Answer.The POI is (2, 6).

 6. y = 1 - x

y = 4 - 4x

Answer.The POI is (1, 0).

 

Use the following descriptions to develop two equations. Then graph the equations to solve them simultaneously:

7. The sum of two numbers is 19 and the difference between those numbers is 5.

Find the value of those two numbers.

Hint.Let the numbers be x and y.
Then x + y = 19 and x - y = 5.

Answer.The numbers are 12 and 7.

 8. Three biscuits and 4 cakes cost 55 cents while the cost of two biscuits and 5 cakes cost 60 cents.

Find the cost of each.

Hint.Let the biscuits be b and the cakes be c.
Then 3b + 4c = 55
and 2b + 5C = 60.

Answer.The biscuits cost 5c each
and the cakes cost 10c each.
9. Two customers enter a butcher's shop.

One customer bought 4 kg of steak and 2 kg of sausages and paid $136.

The second customer bought 1 kg steak and 5 kg of sausages and paid $88.

What were the prices per kilogram for the steak and the sausages?

Answer.Steak: $28/kg.
Sausages: $12/kg.

 10.

Write the cost and revenue equations for each of the following situations and then graph the equations to answer the accompanying questions:

11. Tom provides lawn mowing and garden services. He has overheads of $135 per week (insurance, depreciation on equipment, registration, etc) and it costs him $35 per hour for his overheads (petrol, replacement parts, etc).

Tom charges his customers at $50 per hour.

(i) After how many hours do Tom's income equal his costs?

(ii) What is Tom's financial position for the week if he works for only 20 hours?

Answer.(i) Breakeven at 9 hours.
(ii) He has $165 profit (small business struggles!).

 12.
   
   

Answer the following questions related to the graph given for that question.

17.

The linear graphs at the left show the cost of making a sandwich and the income received from selling the sandwiches.

(i) Let the income received be $I and n be the number of sandwiches sold. Write a formula for the income.

(ii) Let the costs of making a sandwich be $C and n be the number of sandwiches sold. Write a formula for the costs.

(iii) How many sanwdiches have to be sold to break-even?

(iv) What is the profit if 7 sandwiches are sold?
18. A small company assembles mobile phones for use in villages. Those in charge of the company use the following equations to model:
  • the cost ($C) of making n phones: C = 6,000 + 300n
  • the income $I received from selling the n phones: I = 500n

 

Use the two equations to answer the following:

(i) What are the fixed costs if no phones are made?

(ii) What is the income from selling 2,000 phones?

(iii) What is the cost of assembling those 2,000 phones?

(iv) What profit does the company make if 2,000 phones are sold?

(v) What is the minimum number of phones that must be sold if the company just covers costs?
(you may wish to graph the two equations to answer this question or you could solve the two equations simultaneously by equating the right hand sides as both equations determine the money involved).