So,
Let the amount of the expensive coffee beans be x, and the amount of the less expensive coffee bean be y.
The number of pounds of coffee beans is 50.
x + y = 50
$8 times the amount of more expensive coffee beans plus $4 times the amount of less expensive coffee beans divided by 50 equals $6.50.
[tex] \frac{8x + 4y}{50} = 6.50[/tex]
Now, all we have to do is use Elimination by Substitution.
x + y = 50
Subtract y from both sides.
x = 50 - y
Substitute 50 - y for x in the second equation.
[tex]\frac{8(50 - y) + 4y}{50} = 6.50[/tex]
Distribute.
[tex]\frac{400 - 8y + 4y}{50} = 6.50[/tex]
Collect Like Terms.
[tex]\frac{400 -4y}{50} = 6.50[/tex]
Simplify.
[tex]\frac{4(100 -y)}{2*5^2} = 6.50[/tex]
[tex]\frac{2(100 -y)}{5^2} = 6.50[/tex]
[tex]\frac{200 - 2y}{25} = 6.50[/tex]
Multiply both sides by 25.
[tex]200 - 2y = 162.5[/tex]
Add 2y to both sides.
[tex]200 = 162.5 + 2y[/tex]
Subtract 162.5 from both sides.
37.5 = 2y
Divide both sides by 2.
18.75 = y
Substitute in an earlier equation to solve for x.
x = 50 - y
x = 50 - 18.75
x = 31.25
Check.
31.25 + 18.75 = 50 lbs.
[tex] \frac{8x + 4y}{50} = 6.50[/tex]
[tex] \frac{8(31.25) + 4(18.75)}{50} = 6.50[/tex]
[tex] \frac{250 + 75}{50} = 6.50[/tex]
[tex] \frac{325}{50} = 6.50[/tex]
[tex] \frac{5^2*13}{2 * 5^2} = 6.50[/tex]
[tex] \frac{13}{2} = 6.50[/tex]
6.50 = 6.50 This also checks.
There were 31.75 pounds of the more expensive coffee bean and 18.75 pounds of the less expensive coffee bean.
