systems of linear equations

Question

Josie wants to make 10 pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be $54. Nuts cost $6 per pound, and raisins cost $3 per pound. Solve the system

{n+r=106n+3r=54

Answer 1

The actual question here is How many pounds of each ingredient must she use to both end up with 10 lbs and meet her budget.

The first equation is the easy one.

let n = the number of pounds of nuts

let r = the number of pounds of raisins

We know that the finished product must weight 10 lbs.

Therefore, n + r = 10

The other equation deals with the cost of the ingredients.

We know that the total cost must be $54. That, of course, is the combined cost of the nuts and raisins.

The cost of the nuts is $6 per pound times the number of pounds, so 6n

The cost of the raisins is $3 per pound times the number of pounds, so 3r

Thus, the second equation is 6n + 3r = 54

Now we use substitution to begin solving

n + r = 10

n = 10 - r (subtract r from both sides)

We now take this expression for n and plug it into the other equation:

6n + 3 r = 54

6(10-r) + 3 r = 54

60 - 6r + 3r = 54 (subt 54 and add 3r to both sides)

6 = 3r

2 = r

If n + r = 10 and r = 2, then n = 8

Check by plugging these answers into the other formula

Answer 2

To solve this system of equations we will use the method of substitution.

Take the equation n+r=10. This is the easiest to work with. We can solve this equation for either n or r. Let's solve for n.

n+r=10 (subtract r from both sides)

n=10-r

Now we will substitute (10-r) into the second equation, 6n+3r=54, everywhere we see an n.

6n+3r=54

6(10-r)+3r=54 (now distribute)

60-6r+3r=54 (combine like terms)

-3r=-6 (divide both sides by -3)

r=2

Josie will want 2 pounds of raisins. To find the number of pounds of nuts let's re-visit our first equation: n=10-r. Simply substitute r=2 in the equation to get n=10-2=8. So Josie will use 8 pounds of nuts along with her 2 pounds of raisins.

Best wishes!