Use Gaussian elimination or Gauss–Jordan elimination....

Use Gaussian elimination or Gauss–Jordan elimination. Ishikawa Manufacturing borrowed $\$ 30,000$ to buy a new piece of equipment. Part of the money was borrowed at $8 \%$, part at $10 \%$, and part at $12 \% .$ The annual interest was $\$ 3040,$ and the total amount borrowed at $8 \%$ and $10 \%$ was twice the amount borrowed at $12 \% .$ How much was borrowed at each rate?

Use Gaussian elimination or Gauss–Jordan elimination.
Ishikawa Manufacturing borrowed $\$ 30,000$ to buy a new piece of equipment. Part of the money was borrowed at $8 \%$, part at $10 \%$, and part at $12 \% .$ The annual interest was $\$ 3040,$ and the total amount borrowed at $8 \%$ and $10 \%$ was twice the amount borrowed at $12 \% .$ How much was borrowed at each rate?

Key Concepts

Gauss–Jordan Elimination

Gauss–Jordan elimination is an extension of Gaussian elimination that further reduces the augmented matrix to reduced row echelon form. This method eliminates variables one by one until each leading coefficient is 1 and is the only non-zero entry in its column. It provides a more direct way of reading off the solutions of the system and can help in verifying the uniqueness or dependency of solutions.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves applying a sequence of row operations to transform the system’s augmented matrix into an upper triangular form, from which back-substitution can be used to determine the solution. This technique is a fundamental tool in linear algebra.

Translation of Word Problems to Algebraic Equations

This concept involves reading a real-world scenario and then identifying the unknown quantities and the relationships among them to create mathematical equations. It is a critical problem-solving skill that bridges verbal descriptions and mathematical formulations, enabling the use of algebraic methods to find solutions.

Interest Rate Calculations

Interest rate calculations pertain to determining the amount of interest earned or paid based on the principal amount, the interest rate, and the time period involved. In financial contexts, these calculations often require setting up equations that relate rates and amounts, which is essential for analyzing loans, investments, and other economic activities.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. In many real-world problems, different conditions are represented by separate linear equations, and the solution to the system is the set of values that satisfies all the given equations simultaneously. Understanding how to set up and solve these systems is crucial in fields ranging from engineering to economics.

Transcript

Please give Ace some feedback