# APPLICATIONS OF LINEAR EQUATIONS

APPLICATIONS OF LINEAR EQUATIONS

The Cartesian coordinate system also known as the rectangular coordinate system is used widely in business, science and methematics to show a relationship between two variables. One variable is represented along the horizontal axis or X-axis and the other variable is represented along the vertical axis or Y-axis. A linear relationship between the variables is represented on the coordinate system as a straight line.

Illustration Number One :

A company purchases a computer system for $10, 000. Figure One is a graph that shows the depreciation of the computer system over a five-year period. Information can be obtained from the graph. For example, the depreciated value of the computer after two years is $6, 000; the depreciated value of the computer after four years is $2,000.

From the graph, an equation of the line that represents the depreciation can be derived.

First, use any two points shown on the graph to find the slope of the line. Note that the points on the graph are as follows: (0, 10,000) ; (1, 8000); (2, 6000); (3, 4000); (4, 2000) ; (5, 0).

Consider (0, 10,000) and ( 5, 0) to get the value of the slope.

(X1, Y1) = (0, 10,000) (X2, Y2) = (5, 0)

m (slope) = (Y2 - Y1)/(X2 - X1)

m = (0 - 10, 000 )/ (5 - 0 ) = -10, 000/5 = -2,000

m = -2000

Then locate the Y-intercept of the line on the graph. The Y-intercept is the point on the Y-axis where X = 0. The Y-intercept is at (0, 10,000).

Finally, we use the slope-intercept form of an equation to write the equation of the line.

Y = mX + b

Y = -2000X + 10, 000

This is the equation of the line which represents depreciation.

In the equation , the slope represents the annual depreciation of the computer ,The Y-intercept represents the computer's value at the time of purchase . Once the equation of the line has been established, the equation can be used to determine the depreciated value of the computer system after any given number of years.

To find the depreciated value of the computer after two and one-half years, substitute 2.5 for X in the equation and solve for Y,

Y = -2000X + 10, 000

Y = (-2000) (2.5) + 10, 000

Y = -5,000 + 10, 0000

Y = 5, 000

The depreciated value of the computer system after two and a half years is $5, 000.

Illlustration Two :

Figure Two is a graph that shows the relationship between the total cost of manufacturing toasters and the number of toasters manufactured. Write the equation of the line that represents the total cost of manufacturing the toasters. Use the equation to find the total cost of manufacturing 700 toasters.

Solution :

First, establish the equation. Use any two points on the graph to find the slope of the line.

(X1, Y1) = ( 0, 1,000)

(X2, Y2) = (500, 5000)

m = (Y2 - Y1)/(X2 - X1) = (5,000 - 1,000)/(500 - 0)

m = 4, 000/ 500 = 8

The Y-intercept is at ( 0, 1000)

The equation therefore is Y = 8X + 1000

In the equation, the slope represents the unit cost , or the cost to manufacture one toaster. The Y- intercept represents the fixed costs of operating the plant. To find the cost of manufacturing 700 toasters, substitute 700 for X:

Y = 8 (700) + 1000

= 5, 600 + 1000 = $6, 600

The cost of manufacturing 700 toasters is $6, 600.

SOURCE : INTERMEDIATE ALGEBRA

AN APPLIED APPROACH

THIRD EDITION By

Aufmann/Barker

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