Answer
The answer is: 8 boards at 6.50 dollars and 12 boards at 9.50 dollars.
Work Step by Step
We assign the variables:
x = size #1
y = seze #2
Step 1:
Find the equations that represent the problem.
We know that $x+y=20$ because the addition of the amount boards of both sizes has to equal the total 20 boards.
We solve for x to make step 2 easier.
$x=20-y$ -> Eq. 1
We can get the second equation from the wording in the problem.
From "The total cost of 20 boards is $\$166$. One size costs $\$6.50$, and the second size costs $\$9.50$", we get
$6.5x+9.5y=166$ -> Eq. 2
Step 2:
Solve the system of equations. Using the substitution method,
-> Substitute Eq. 1 into Eq. 2
$6.5(20-y)+9.5y=166$
$130-6.5y-9.5y=166$
$3y=166-130$
$3y=36$
$y=\frac{36}{3}$
$y=12$
-> Substitute the value for $y$ into Eq. 1
$x=20-12$
$x=8$
Step 3:
The answer is: 8 boards at 6.50 dollars and 12 boards at 9.50 dollars.
