Acompany sells herb blends for cooking. the tables below show the number of teaspoons of each herb in each blend and the cost for each blend.

number of teaspoons of herb per blend

basil
oregano
parsley
blend 1
1
2
3
blend 2
1
1
2
blend 3
2
2
1

cost
blend 1
$1.06
blend 2
$0.72
blend 3
$0.99

what is the cost of basil per teaspoon?
$0.15
$0.18
$0.19
$0.23

  $0.23

Step-by-step explanation:

You can subtract the cost of Blend 3 from double the cost of Blend 2 and you will find the value of Parsley:

  2(1 tsp Basil + 1 tsp Oregano + 2 tsp Parsley) - (2 tsp Basil + 2 tsp Oregano + 1 tsp Parsley) = 2($0.72) - ($0.99)

  3 tsp Parsley = $0.45 . . . . . simplify

Dividing by 3, we find

  1 tsp Parsley = $0.15

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Now we can subtract the cost of Blend 1 from double the cost of Blend 2 and subtract another tsp of Parsley to find the cost of Basil.

  2(1 tsp Basil + 1 tsp Oregano + 2 tsp Parsley) - (1 tsp Basil + 2 tsp Oregano + 3 tsp Parsley) - (1 tsp Parsley) = 2($0.72) - ($1.06) - ($0.15)

  1 tsp Basil = $0.23 . . . . . simplify

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I find it easier to work systems of linear equations like this using a calculator capable of "row reduction". Each equation of the system corresponds to one row of an "augmented matrix." Input to an on-line calculator is shown in the first attachment; its output is shown in the second attachment. Since the price of Basil is the first variable, its value of 0.23 per tsp is the value the question is asking for.

Basically, reduction of the augmented matrix to row-echelon form is a systematic way to solve the system by elimination. Since we were only looking for the cost of Basil, we didn't have to solve the entire system.

Had we used Cramer's method, we would have arrived at the solution by evaluating the determinants of two 3×3 matrices. The method shown above uses fewer calculations. (It also requires a certain amount of insight or "trial and error" to find an efficient way to get the answer to the question.)

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