Exercise 9 Page 436

There are two dry cleaning companies that offer a home pick-up and delivery service. Prices are shown in the table below.

We can write the equations for the costs of each company in slope-intercept form. y= mx+ b In this form, m is the slope and b is the y-intercept. We will find each slope by using the Slope Formula. m=y_2-y_1/x_2-x_1 In the above formula, (x_1,y_1) and (x_2,y_2) represent pairs of data entries that satisfy the equation.

Equation for Company 1

Let's start with the equation for Company 1, y=A(x). In order to determine its slope, we will use the coordinate pairs (5,55.25) and (10,75.50).

Thus, the slope for A(x) is 4.05. A(x)= 4.05x+ b We will now substitute the point (5,55.25) in the above equation in order to determine the y-intercept b.

The y-intercept is 35. With this, we have everything we need to form an equation for the cost of the service from Company 1. A(x)= 4.05x+ 35

Equation for Company 2

We will now write an equation, y=B(x), for Company 2. Let's use (5,31.25) and (10,57.50) as our coordinate pairs to determine the slope.

The slope of B(x) is 5.25. B(x)= 5.25x+ b Let's find the y-intercept by substituting the point (5,31.25) in the above equation.

We found the y-intercept to be 5 and thus we can write our second equation. B(x)= 5.25x+ 5

System of Equations

We will write a system of linear equations using the equations we have written above. A(x)=4.05x+35 & (I) B(x)=5.25x+5 & (II) We want to determine the number of garments, x, for which both companies charge the same. To do so, we need to solve the equation A(x)=B(x). A(x)&=B(x) 4.05x+35&=5.25x+5 Let's use inverse operations to isolate the x-variable.

Both companies charge the same amount for cleaning 25 garments.

Calculating the Price

We can find the cost of cleaning 25 garments by substituting 25 for x in either of the equations. Let's do this using A(x).

Both companies charge $136.25 for cleaning 25 garments.