Question

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 silver (of specific gravity 10 ) and \(y \mathrm{gm}\) of gold (of specific gravity 20 ) weighs \(120 \mathrm{gm}\) in air and \(109 \mathrm{gm}\) in water. Find \(x\) and \(y\).

Short answer

x = 20, y = 100.

Step-by-step solution

Identify the system of equations

First, set up two equations based on the problem's conditions. The first equation comes from the total weight in the air, and the second equation comes from the effective weight in water.

Equation for total weight in air

Since the total weight of the object is the sum of the weights of silver and gold and is 120 gm in air, write this equation as: x + y = 120

Equation for effective weight in water

The effective weight in water is the total weight minus the buoyant force. The buoyant force is equal to the volume of the displaced water, which depends on the specific gravity of silver and gold. The equation considering this can be written as: x/10 + y/20 = 11 This follows because 109 gm in water means a buoyant force of 120 - 109 = 11 gm.

Set up the system of equations

The system of equations to solve is: x + y = 120 x/10 + y/20 = 11

Convert to standard form

Rewrite the second equation to avoid fractions by multiplying through by 20: x + 2y = 220

Row reduction method

Arrange the augmented matrix from the equations: [ 1 1 | 120 ] [ 1 2 | 220 ] Subtract the first row from the second row: [ 1 1 | 120 ] [ 0 1 | 100 ] Use the second row to simplify the first row: [ 1 0 | 20 ] [ 0 1 | 100 ]

Solution

From the row-reduced matrix, the solutions are: x = 20 y = 100

Key concepts

system of equations

When dealing with real-world problems, it's often necessary to determine the values of unknowns that satisfy multiple conditions simultaneously. A system of equations is a set of two or more equations that have common variables. In our example, we have two unknowns, \(x\) (the weight of silver) and \(y\) (the weight of gold), and two conditions: the total weight in air and the effective weight in water. These conditions are represented by two equations:

• Total weight in air: \(x + y = 120\)
• Effective weight in water: \(\frac{x}{10} + \frac{y}{20} = 11\)

By solving these equations together, we can find the specific values of \(x\) and \(y\) that meet both criteria. This process often involves methods like substitution, elimination, or matrix techniques.

determinants

Determinants are a useful tool in linear algebra, especially when solving systems of linear equations. They are calculated from a square matrix and can indicate many properties of the matrix. For a 2x2 system, the determinant \(D\) of a matrix\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \] is found using the formula:
\[ D = ad - bc \]

In our example, the system of equations can be represented in matrix form as:
\[ \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 120 \ 220 \end{pmatrix} \]

The determinant of the coefficient matrix \( \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix} \) is \(1(2) - 1(1) = 1\).

Using determinants, we can solve systems directly by finding the inverse of the coefficient matrix or using Cramer's rule.

row reduction

Row reduction, also known as Gaussian elimination, is a method of solving systems of linear equations by transforming the system's augmented matrix into a simpler form. The goal is to convert the matrix into row-echelon form and then into reduced row-echelon form (RREF).

For our example:

• Start with the augmented matrix: \( \begin{pmatrix} 1 & 1 & 120 \ 1 & 2 & 220 \end{pmatrix} \)
• To eliminate \(x\) from the second row, subtract the first row from the second: \( \begin{pmatrix} 1 & 1 & 120 \ 0 & 1 & 100 \end{pmatrix} \)
• To simplify further, use the second row to clear the coefficient of \(y\) in the first row: \( \begin{pmatrix} 1 & 0 & 20 \ 0 & 1 & 100 \end{pmatrix} \)

From this, we see directly that \(x = 20\) and \(y = 100\). Row reduction is a step-by-step process that systematically uses row operations to make the solving process easier.