Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Malia purchased 1 share in the biotech firm for $48. Patrick i... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Malia purchased 1 share in the biotech firm for $48. Patrick invested $3,144 in 90 shares in the software company and 43 shares in the biotech firm. How much did each share cost?
Understand the Problem
The question is asking for a system of equations to be formulated based on the given investment scenario involving shares in two companies, and then to solve it using an augmented matrix to determine the costs of individual shares.
Answer
Each share in the software company costs $12, and each share in the biotech firm costs $48.
Answer for screen readers
Each share in the software company costs $12, and each share in the biotech firm costs $48.
Steps to Solve
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Define Variables Let ( x ) be the cost of each share in the software company, and ( y ) be the cost of each share in the biotech company.
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Set Up Equations From the problem:
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Malia bought 1 share of the biotech company for $48: $$ y = 48 $$
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Patrick invested a total of $3,144 in 90 shares of the software company and 43 shares of the biotech company. This can be expressed as: $$ 90x + 43y = 3144 $$
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Create the Augmented Matrix The system of equations can be represented in augmented matrix form: $$ \begin{bmatrix} 0 & 1 & | & 48 \ 90 & 43 & | & 3144 \end{bmatrix} $$
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Use Row Reduction Perform row operations to reduce the matrix to row-echelon form. We aim to eliminate variables:
- Use the first equation to eliminate ( y ) in the second equation.
- Solve for Variables After row reduction, solve for ( x ) and ( y ) by back substitution. Substitute ( y = 48 ) into the second equation: $$ 90x + 43(48) = 3144 $$
Calculate ( 43 \times 48 ): $$ 43 \times 48 = 2064 $$
Now solve for ( x ): $$ 90x + 2064 = 3144 \ 90x = 3144 - 2064 \ 90x = 1080 \ x = \frac{1080}{90} \ x = 12 $$
- Final Prices Now we have:
- Cost of the software share ( x = 12 )
- Cost of the biotech share ( y = 48 )
Each share in the software company costs $12, and each share in the biotech firm costs $48.
More Information
In this scenario, it highlights how to effectively use systems of equations to solve real-world problems involving investment. Understanding the setup of the equations is crucial for solving such problems.
Tips
- Mixing up the variables: It's important to clearly define what each variable represents (e.g., which corresponds to software and which to biotech).
- Miscalculating while solving the augmented matrix or during row operations. Always double-check calculations during each step.
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