Solving Linear Equations for Income and Expenditure Ratios

Problem Statement and Mathematical Context

Determining individual incomes and expenditures from given ratios and fixed savings is a classic application of simultaneous linear equations in two variables. In this problem setting, two individuals, $A$ and $B$, have monthly incomes that exist in a fixed ratio, as well as monthly expenditures that exist in another distinct ratio. Both individuals save the exact same amount of money each month.

The ratio of the monthly incomes of person $A$ to person $B$ is given as $8 : 7$

The ratio of the monthly expenditures of person $A$ to person $B$ is given as $19 : 16$

Each individual achieves a monthly savings of $\text{₹ } 2550$

Algebraic Formulation of Linear Equations

To convert the proportional relationships into solvable algebraic equations, scalar multipliers must be defined for income and expenditure respectively.

Let $x$ represent the common scaling factor for income. Consequently, the monthly income of person $A$ is represented as $8x$, and the monthly income of person $B$ is represented as $7x$

Let $y$ represent the common scaling factor for expenditure. Consequently, the monthly expenditure of person $A$ is represented as $19y$, and the monthly expenditure of person $B$ is represented as $16y$

The fundamental economic relationship connecting income, expenditure, and savings is given by the formula:

$\text{Income} - \text{Expenditure} = \text{Savings}$

Applying this fundamental relation to person $A$ yields the first linear equation:

$8x - 19y = 2550$

Applying this fundamental relation to person $B$ yields the second linear equation:

$7x - 16y = 2550$

Comprehensive Step-by-Step Solution Methods

The pair of linear equations in two variables can be solved using either the Method of Elimination or the Method of Substitution / Equating Expressions.

Method 1: Simultaneous Elimination

Consider the system of equations:

$8x - 19y = 2550$

$7x - 16y = 2550$

To eliminate the variable $x$, multiply the first equation by $7$ and the second equation by $8$

$7 \times (8x - 19y) = 7 \times 2550$

$56x - 133y = 17850$

$8 \times (7x - 16y) = 8 \times 2550$

$56x - 128y = 20400$

Subtracting the expanded first equation from the expanded second equation cancels the $56x$ term:

$(56x - 128y) - (56x - 133y) = 20400 - 17850$

$5y = 2550$

$y = \frac{2550}{5}$

$y = 510$

Substituting the value $y = 510$ back into the first equation $8x - 19y = 2550$

$8x - 19(510) = 2550$

$8x - 9690 = 2550$

$8x = 2550 + 9690$

$8x = 12240$

$x = \frac{12240}{8}$

$x = 1530$

Method 2: Equating Savings Expressions

Since both individuals save an identical amount of $\text{₹ } 2550$, the expression for savings for person $A$ can be equated directly to the expression for savings for person $B$

$8x - 19y = 7x - 16y$

Rearranging the linear equation to isolate $x$ in terms of $y$

$8x - 7x = 19y - 16y$

$x = 3y$

Substituting $x = 3y$ into the equation for person $A$

$8(3y) - 19y = 2550$

$24y - 19y = 2550$

$5y = 2550$

$y = 510$

Using $x = 3y$ to determine $x$

$x = 3 \times 510$

$x = 1530$

Final Values and Verification of Results

With the unknown multipliers $x = 1530$ and $y = 510$ fully resolved, the exact monthly income and expenditure values for both individuals can be explicitly calculated.

The monthly salary or income of person $A$ is:

$\text{Salary of } A = 8x = 8 \times 1530 = \text{₹ } 12240$

The monthly salary or income of person $B$ is:

$\text{Salary of } B = 7x = 7 \times 1530 = \text{₹ } 10710$

To verify the consistency of these answers against expenditure and savings:

The monthly expenditure of person $A$ is $19y = 19 \times 510 = \text{₹ } 9690$. Subtracting expenditure from income yields $\text{₹ } 12240 - \text{₹ } 9690 = \text{₹ } 2550$, which matches the given monthly savings.

The monthly expenditure of person $B$ is $16y = 16 \times 510 = \text{₹ } 8160$. Subtracting expenditure from income yields $\text{₹ } 10710 - \text{₹ } 8160 = \text{₹ } 2550$, which also matches the given monthly savings.