Pair of Linear Equations in Two Variables: Detailed Graphical and Algebraic Problems

Graphical Representations of Linear Systems

The study of linear equations in two variables begins with graphical analysis to determine solutions and geometric properties. One specific exercise requires drawing the graphs for the equations $2x + y - 10 = 0$ and $x - y = -1$. This process involves plotting the lines on a Cartesian plane to identify the coordinates of the vertices of the triangle formed by these two straight lines in conjunction with the x-axis. Additionally, the task requires calculating the area of this specific triangular region. Another problem involves graphing the pair of equations defined as $x + 2y = 5$ and $2x - 3y z - 4$. The objective is to determine the exact points where these lines intersect the x-axis.

Graphical methods are also applied to determine the nature of solutions for a given system. For the equations $2x - 3y = 5$ and $3x + 4y = -1$, one must determine graphically if the system possess (i) a unique solution, (ii) infinitely many solutions, or (iüi) no solution. Furthermore, the system comprising $2x + y = 6$ and $2 - y + 2 = 0$ must be solved graphically. In this scenario, a secondary objective is to find the ratio of the areas of the two triangles formed by the lines representing these equations: one triangle formed with the x-axis and the other formed with the y-axis. The provided answer for this area ratio is $4 : 1$.

Problems Pertaining to Age Relations

Algebraic modeling is frequently used to resolve complex age-related questions. In one instance, a father's age is described as thrice the sum of the ages of his two children. It is stated that after a period of $5$ years, the father's age will transition to being exactly twice the sum of their ages at that time. The solution identifies the father's age as $45$ and the sum of the children's ages as $15$. In a separate case involving Salim and his daughter, it was noted that $2$ years ago, Salim was thrice as old as his daughter. Furthermore, $6$ years into the future, Salim will be $4$ years older than twice her age. Their present ages are determined to be $38$ for Salim and $14$ for his daughter.

Additional age problems include a scenario where a father is currently $3$ times as old as his son, and in $12$ years, he will be twice as old as his son. The current ages for the father and son are $36$ and $12$ respectively. Another problem examines the relationship between a mother and her daughter; $5$ years from now, the mother's age will be $3$ times the age of her daughter. Conversely, $5$ years ago, the mother was $7$ times as old as her daughter. Their present ages are calculated as $40$ and $10$.

Numerical and Digit-Based Algebraic Problems

Digit-based problems require an understanding of place value and reversing digit order. For a two-digit number where the sum of the digits is $15$, it is observed that reversing the order of the digits produces a number that exceeds the original given number by $9$. The specific number for this problem is $78$. Another problem considers a two-digit number where the sum of the number itself and the number obtained by reversing its digits is $99$. If the digits have a difference of $3$, the problem suggests discussing two distinct cases: where the tens digit $x$ is greater than the units digit $y$ ($x > y$) and where the tens digit is less than the units digit ($x < y$). The resulting possible numbers are $63$ and $36$.

Further examples include a two-digit number that is exactly $4$ times the sum of its constituent digits. If the value $18$ is added to the number, the digits are reversed. The solution for this number is $24$. Additionally, a complex digit problem states that $8$ times a two-digit number is equal to $3$ times the number formed by reversing the order of its digits. Given that the difference between the digits of the number is $5$, the corresponding number is identified as $27$.

Rational Expressions and Fraction-Based Scenarios

Problems involving fractions involve manipulating numerators ($x$) and denominators ($y$) based on given conditions. In one problem, the sum of the numerator and the denominator is described as being $4$ more than twice the numerator. When $3$ is added to both the numerator and the denominator, the resulting values form a ratio of $2 : 3$. The sought fraction is documented as $\frac{5}{9}$. A similar case states that the sum of the numerator and denominator is $12$. If the denominator is increased by a value of $3$, the fraction changes, and the final answer is determined to be $\frac{5}{7}$.

A third fraction problem specifies that the sum of the numerator and denominator is $3$ less than $2$ the denominator (implied twice the denominator). If both the numerator and denominator are decreased by $1$, the numerator becomes the denominator. This specifically formulated scenario results in the fraction $\frac{4}{7}$. Finally, another problem repeats the condition where the sum of the numerator and denominator is $4$ more than twice the numerator, and an increase of $3$ to both parts results in a ratio of $2 : 3$, confirming the fraction as $\frac{5}{9}$.

Geometric Transformations and Cost Analysis Models

Geometric problems focus on how changes in dimensions affect the area of shapes. For a rectangle, if the length is increased and the breadth is reduced by $2$ units each, the area suffers a reduction of $28 \text{ sq.units}$. However, if the length is reduced by $1$ unit and the breadth is increased by $2$ units, the overall area increases by $33 \text{ sq.units}$. The dimensions for this rectangle are found to be $23$ and $11$, with a total area of $253 \text{ sq.units}$. In another rectangle problem, the area remains the same if the length increases by $7 \text{ m}$ and the breadth decreases by $3 \text{ m}$. The area also remains unaffected if the length is decreased by $7 \text{ m}$ and the breadth is increased by $5 \text{ m}$. The dimensions for this specific rectangle are $28$ and $15$.

Economic and cost analysis problems are also explored through linear systems. Monthly household expenditures are modeled by a fixed rent and a variable mess charge per person. For $2$ people, the expenditure is $3900$, and for $5$ people, it is $7500$. The rent charges are determined to be $1500$, while the mess charges per head per month are $1200$. Another problem involves the cost of furniture: $4$ tables and $3$ chairs cost $2250$, while $3$ tables and $4$ chairs cost $R1950$. The calculation for the cost of $8$ chairs results in $1200$ and for $10$ tables results in $4500$.

Finally, a distribution problem between two individuals, A and B, involves mangoes. A states to B that if B gives A $30$ mangoes, A will have twice as many as remain with B. B counters that if A gives B $10$ mangoes, B will have thrice as many as remain with A. The number of mangoes held by A is $34$ and by B is $62$.