Q:

Venetta buys 2 pounds of pistachios and 3 pounds of almonds. The pistachios cost $4 more per pound than the almonds. She pays a total of $48. Which of the following are true? Select all that apply. A. One pound of pistachios plus 1 pound of almonds cost $20. B. The pistachios cost twice as much per pound as the almonds. C. Reducing the number of pounds of almonds by one results in a total cost of $40. D. The cost a, in dollars, of 1 pound of almonds is modeled by 2(a – 4) + 3a = 48. E. The cost p, in dollars, of 1 pound of pistachios is modeled by 2p + 3(p – 4) = 48.

Accepted Solution

A:
We want to find a system of equations, and by solving that system we will be able to see which statements are true.We will see that options A, C, and E are true.We know that:Venetta buys 2lb of pistachiosVenetta buys 3 lb of almonds.Let's define the variables:x = price per pound of pistachiosy = price per pound of almonds.Now we also know that:"The pistachios cost $4 more per pound than the almonds."This can be written as:[tex]x = y + \$4[/tex]"She pays a total of $48"This can be written as:[tex]2*x + 3*y = \$48.[/tex]So a system of equations:[tex]x = y + \$4[/tex][tex]2*x + 3*y = \$48.[/tex]To solve this system, the first thing we need to do is isolate one of the variables in one of the equations, particularly we can see that x is already isolated in the first equation, so we can skip that step.Now we can replace the isolated variable in the other equation to get:[tex]2*(y + \$4) + 3*y = \$48[/tex]now we can solve this for y:[tex]2*y + \$8 + 3*y = \$48[/tex][tex]5*y + \$8 = \$48[/tex][tex]5*y = \$48 - \$8 = \$40[/tex][tex]y = \$40/5 = \$8[/tex]now that we know this, we can use:[tex]x = y + \$4 = \$8 + \$4 = \$12[/tex]now that we know:y = $8x = $12Let's see which statements are true:A) One pound of pistachios plus 1 pound of almonds cost $20.True, $8 + $12 = $20.B) Β The pistachios cost twice as much per pound as the almonds.False, $12 is not the double of $8.C) Reducing the number of pounds of almonds by one results in a total cost of $40. True, one pound less of almonds means $8 less in the price.D) The cost a, in dollars, of 1 pound of almonds is modeled by 2(a – 4) + 3a = 48Simplifying the expression we get:2*(a - 4) + 3a = -8 + a = 48a = 48 + 8 = 52This clearly does not model the price of one pound of almonds, this statement is false.E) The cost p, in dollars, of 1 pound of pistachios is modeled by 2p + 3(p – 4) = 48.Solving the equation we get:2*p + 3*p - 12 = 485*p = 48 + 12 = 60p = 60/5 = 12This is true.If you want to learn more, you can read: