U4D5 Systems of Linear Equations: Mixture and Percentage Applications

Vocabulary and core modeling concepts for mixture, percentage, investment, and sequence applications using systems of linear equations.

Four-Step Procedure for Systems Modeling

The standard process for solving applied linear systems: a) Introduce variables to represent the unknown values, b) Form a system of equations involving the variables, c) Solve the system, and d) Answer the problem and check the solution.

Quantity Equation (Mixture Applications)

The linear equation representing the sum of individual masses, weights, or liquid volumes of the components equaling the total amount of the combined mixture (e.g., $x + y = 50\,\text{kg}$).

Value Equation (Mixture Applications)

The linear equation equating the sum of the monetary values or pure solute quantities of each component to the total value or content of the final mixture (e.g., $c_1 x + c_2 y = C_{\text{final}} \times \text{total}$).

Carat (Gold Purity)

A fractional measure of gold purity based on parts per $24$, given by $\frac{\text{carats}}{24}$, where pure gold is $24$-carat ($100\%$), $12$-carat is $\frac{12}{24}$ ($50\%$), and $21$-carat is $\frac{21}{24}$ ($87.5\%$).

Annual Interest Equation (Percentage Applications)

A linear relation expressing total interest gained or lost from multiple accounts as the sum of each principal multiplied by its annual interest rate in decimal form (e.g., $0.105x + 0.12y = 5520$).

Concentration by Volume

The proportion of pure solute present in a liquid solution, where the total volume of pure substance is calculated by multiplying the decimal percentage by the total solution volume (e.g., $60\%$ of $180\,\text{mL}$ is $0.60 \times 180 = 108\,\text{mL}$ of pure $\text{HCl}$).

Income-Expenditure-Savings Relation

The algebraic model defined by $\text{Income} - \text{Expenditure} = \text{Savings}$, used to construct linear systems when percentage changes alter income, expenses, and net savings between years.

Quadratic Height Model for Bridge Rods

The formula $h = a + bn^2$, relating the vertical height $h$ (in metres) of the $n^{\text{th}}$ suspension rod from the centre to constants $a$ and $b$, which are solved using a system formed from consecutive sequence terms.