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The linear equation for Ford Motors in the form Y1=mX1 +b is given as;

Y1= 350X1

+ 320.

On the other hand, the linear equation for Toyota Motors in the form Y2=mX2 +b is given by;

Y2= 210X2

+ 2000.

The (x,y) values for the two sets of equations are given as follows;

X1

Y1

X2

Y2

0

320

0

2000

2

1020

2

2420

4

1720

4

2840

6

2420

6

3260

8

3120

8

3680

10

3820

10

4100

12

4520

12

4520

14

5220

14

4940

16

5920

16

5360

18

6620

18

5780

20

7320

20

6200

Using the excel software, the data is represented graphically as shown below;

Mathematically, the point of intersection between the two equations can be calculated as follows;

At the point of intersection, the two equations have the same coordinates, say (x,y).

ð Y1= 350X1 + 320 = Y2= 210X2 + 2000.

But at the point of intersection, X1=X2 =X

350X+ 320 = 210X + 2000

140X= 1680

X=12.

Replacing X=12 in any of the linear equations, we get;

Y= 350(12) + 320 = 4520

ð The point of intersection is (12, 4520).

From the graph, the two lines intersect at point (12, 4520).

The cost for leasing car for the two companies is the same at 12 months. However, Frod Motors offers better deal for a period below 12 months while Toyota Motors offers a better deal after 12 months.

A lease of two years will result to different costs for the two companies as follows;

Frod Motors

Y1= 350(24) + 320 =$ 8720

Toyota Motors

Y2= 210 (24) + 2000 = $ 7040

From the two calculations, Toyota Motors will be the option for a two-year lease.

The difference between the two companies is $ (8720-7040) = $ 1680

Part II

The concept of linear equation is important in analyzing and interpretation the demand and supply of products due to price changes. Given the following information on the demand and supply of pizza;

Quantity demand at zero price= 800 units

Change in demand due to $ 1 price increase= 60 units

Quantity supplied at zero price= 100 units

Change in demand due to $ 1 price increase= 50 units

From the following information, the demand and supply equations are given as

Y1 = -60X1

+ 800 and Y2 = 50X2 + 100

The following data are generated from the two equations;

Price

Demand

Price

Supply

0

800

0

100

2

680

2

200

4

560

4

300

6

440

6

400

8

320

8

500

10

200

10

600

12

80

12

700

14

-40

14

800

16

-160

16

900

18

-280

18

1000

20

-400

20

1100

The graphical representation is shown below,

The point of intersection between the two equations (equilibrium price and quantity) can be obtained mathematically as follows;

At equilibrium, the quantity demanded and supplied are the same.

ð Y1 = -60X1 + 800 = Y2 = 50X2 + 100

-60X1 + 800 =50X2 + 100, but X1=X2=X

-60X + 800 =50X + 100

-60X-50X = 100-800

-110X = -700

X= 6.36

Substituting in one of the equations, Y1 = -60(6.36) + 800 = 418.4

The point of intersection is given as (6.36, 418.4)

From the graph, an increase in the price will reduce the demand while increasing the supply of pizza in the market.

September 18, 2023

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