Write a set of linear equations for each of the following problems and solve them using either determinants or row reduction. An object composed of \(x \mathrm{gm}\) of lead (specific gravity 11\()\) and \(y \mathrm{gm}\) of tin (specific gravity 7 ) weighs \(82 \mathrm{gm}\) in ait and \(77 \mathrm{gm}\) in oil of specific gravity \(\frac{1}{2}\). Find \(x\) and \(y\). .

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear equations. This field forms the foundation for solving many problems in engineering, physics, and computer science. In the context of this exercise, we are dealing with linear equations representing constraints on the amounts of lead and tin in an object. These constraints together form a system which can be solved using techniques from linear algebra.
Using two equations, one representing the weight in air and one accounting for the weight in oil, showcases how linear algebra can be applied to real-world problems. The solution involves manipulating these equations to find the values of the unknown variables, a process that is central to linear algebra concepts.

A system of equations is a set of two or more equations involving the same set of variables. Solving a system of equations means finding the values of these variables that satisfy all equations simultaneously.
In this exercise, we initially set up our system from the problem's conditions:

• The total weight in air: \( x + y = 82 \)
• The weight in oil, considering buoyant forces and specific gravities: \( x \times \frac{1}{11} + y \times \frac{1}{7} + 77 = 82 \)

To find the unknowns \( x \) and \( y \), we transformed the second equation and eventually derived two simplified linear equations. This process is crucial for handling multiple constraints and finding solutions that satisfy all given conditions.

Matrices are rectangular arrays of numbers arranged in rows and columns. They are a fundamental tool in linear algebra for representing and solving systems of linear equations. A key concept is the augmented matrix, which includes the coefficients of the variables and the constants from the equations.
For this problem, the system of equations can be represented in an augmented matrix as:
\[{\begin{bmatrix} 1 & 1 & | & 82 \ 7 & 11 & | & 385 \end{bmatrix}}\]
This matrix helps to apply systematic techniques like row reduction to convert it into row echelon form, from which the solutions can be easily extracted.
Matrices simplify the process of handling multiple linear equations, enabling us to efficiently organize and perform operations to solve for unknown variables.

Row reduction is a method used to simplify matrices to solve systems of linear equations. It involves applying a series of operations to transform the matrix into row echelon form or reduced row echelon form. This process is also called Gaussian elimination.
Here is the procedure applied to our problem's augmented matrix:

• Start with the initial matrix: \[{\begin{bmatrix} 1 & 1 & | & 82 \ 7 & 11 & | & 385 \end{bmatrix}}\]
• Subtract 7 times the first row from the second row to clear the first column of the second row:
  \[{\begin{bmatrix} 1 & 1 & | & 82 \ 0 & 4 & | & 305 \end{bmatrix}}\]

From here, we can easily find \( y \) from the second row and then back-substitute to find \( x \).
Row reduction is an efficient way to handle systems of linear equations, turning them into a form that makes it straightforward to identify the solutions.