Write the system of equations by yourself in order to find the error. Be careful choosing the pairs.
Error: Choosing an incorrect pair that does not exist in the table. System of Equations: A(x)=196x+120 B(x)=206x+10 Containers of Paint: 11
Practice makes perfect
According to the following table a painter can buy 5-gallon containers of paint from two different stores.
Containers of Paint
Company A
Company B
2
$512.00
$422.00
4
$904.00
$834.00
6
$1296.00
$1246.00
8
$1688.00
$1658.00
Now, we will examine the solution of a student. Let's choose the same points that the student chose in order to write the functions in slope-intercept form.
y= mx+ b
In this form, m is the slope and b is the y-intercept. First, let's find the slope by using the Slope Formula.
m=y_2-y_1/x_2-x_1
Here, (x_1,y_1) and (x_2,y_2) represent the pairs that x is the containers of paint and y is the price.
Company A
Let's start with Company A. Let x be the containers of paint and y=A(x) be the price. We will use the pairs (2,512) and (4,904) to find the slope.
We have found the y-intercept as b=120. Thus, the function can be written as the following.
A(x)=196x+120
We can say that the student did not make any mistake while writing the function for Company A.
Company B
Next, we will write a function for Company B in the same way that the student did. However, the student has chosen the pair (2,834) but there is no such a pair. Therefore, we will choose the pairs (4,834) and (6,1246)
We have found the y-intercept as b=10. Thus, the function can be written as the following.
B(x)=206x+10
As a result, the student made a mistake by choosing an incorrect pair while writing the function for Company B.
System of Equations and Verification
Finally, we have the system of equations.
A(x)=196x+120 & (I) B(x)=206x+10 & (II)
Let's solve the system by setting up an equation that A(x) is equal to B(x).
A(x)&=B(x)
196x+120&=206x+10
Let's solve the equation for x.
The solution means that the cost will be the same amount for 11 containers of paint, which is reasonable. Since the slope of B(x) is greater than the slope of A(x), eventually their graphs will intersect. After the intersection point, the price at Company B will be greater.
