Math Problem Statement
A company produces three products every day. Their total production on a certain day is 45 tons. It is found that the production of the third product exceeds the production of the first by 8 tons. The total production of the first and third products is twice the production of the second. Determine the production level of each product using Cramer's rule.
Solution
To solve the problem using Cramer's rule, let's define the variables:
Let:
- be the production of the first product,
- be the production of the second product,
- be the production of the third product.
From the problem, we have three key pieces of information, which can be written as equations:
-
Total production is 45 tons:
-
The production of the third product exceeds the first by 8 tons:
-
The sum of the first and third products' production is twice that of the second:
Rewriting as a system of equations:
We can express the system in a more standard form:
Matrix representation:
This system can be written as:
1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & -2 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 45 \\ 8 \\ 0 \end{pmatrix}$$ ### Using Cramer's rule: Cramer's rule gives the solution of a system $$A\mathbf{x} = \mathbf{b}$$ by computing determinants: - $$x_1 = \frac{\text{det}(A_1)}{\text{det}(A)}$$ - $$x_2 = \frac{\text{det}(A_2)}{\text{det}(A)}$$ - $$x_3 = \frac{\text{det}(A_3)}{\text{det}(A)}$$ Where $$A_1$$, $$A_2$$, and $$A_3$$ are matrices formed by replacing the corresponding columns of $$A$$ with the vector $$\mathbf{b}$$. Let's calculate the solution using Cramer's rule. I'll compute this now.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Matrix Algebra
Cramer's Rule
Formulas
x1 + x2 + x3 = 45
x3 = x1 + 8
x1 + x3 = 2x2
Cramer's rule: x1 = det(A1) / det(A), x2 = det(A2) / det(A), x3 = det(A3) / det(A)
Theorems
Cramer's Rule for solving systems of linear equations
Suitable Grade Level
Grades 10-12