Exercise 8 Page 436

Two online retailers sell organic vanilla extract. Prices are given in the following table.

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 Chef Mate

Let's start with the equation for Chef Mate, y=C(x). In order to determine its slope, we will use the coordinate pairs (2,12.50) and (3,17.25).

Thus, the slope for C(x) is 4.75. C(x)= 4.75x+ b We will now substitute the point (4,22) in the above equation in order to determine the y-intercept b.

The y-intercept is 3. With this, we have everything we need to form an equation for the cost of vanilla extract from Chef Mate. C(x)= 4.75x+ 3

Equation for Grocery Gourmet

We will now write an equation, y=G(x), for Grocery Gourmet. Let's use (2,17) and (3,21) as our coordinate pairs to determine the slope.

The slope of G(x) is 4. G(x)= 4x+ b Let's find the y-intercept by substituting the point (2,17) in the above equation.

We found the y-intercept to be 9 and thus we can write our second equation. G(x)= 4x+ 9

System of Equations

We will write a system of linear equations using the equations we have written above. C(x)=4.75x+3 & (I) G(x)=4x+9 & (II) We want to determine the amount of vanilla extract, x, for which both companies charge the same. To do so, we need to solve the equation C(x)=G(x). C(x)&=G(x) 4.75x+3&=4x+9 Let's use inverse operations to isolate the x-variable.

Both companies charge same amount for 8 ounces of vanilla extract.

Calculating the Price

We can find the price of 8 ounces of vanilla extract by substituting 8 for x in either of the equations. Let's do this using G(x).

Both companies charge $41 for 8 ounces of vanilla extract.