Linear equation

Before understanding the concept can you solve the below problem

1. Garima works at a Reliance Retail gardening supply store in India. At the store, pots are sold for ₹100 each and bags of soil for ₹50 each. A customer named Raj comes in and wants to buy some pots and bags of soil. Garima needs to calculate the total cost for Raj’s purchase. 
2. Arpit is planning a road trip. He works at Tiwari Variety Store, a local shop in India. His car’s fuel efficiency is 30 miles per gallon, and gasoline costs ₹3.50 per liter. Arpit wants to calculate the cost for the gasoline he will need for the trip. 
3. Rohan, an avid cyclist, loves to explore his town. He decides to go for a ride on his bicycle, which travels at a constant speed of 15 miles per hour. Rohan wants to calculate the distance he can cover in a specific amount of time.
4. Syed, a curious student, is learning about temperature conversions. He wants to convert temperatures from Celsius to Fahrenheit using the linear equation
   $F=\frac{9}{5}C+\mathrm{32.<span\; style="color:\; \#222222;\; font-family:\; Verdana,\; BlinkMacSystemFont,\; -apple-system,\; \text{'}Segoe\; UI\text{'},\; Roboto,\; Oxygen,\; Ubuntu,\; Cantarell,\; \text{'}Open\; Sans\text{'},\; \text{'}Helvetica\; Neue\text{'},\; sans-serif;"> </span>}$
5. Shubham Gupta is a diligent teenager who is eager to earn some extra money. He decides to do household chores to earn an allowance. Shubham’s plan is to earn a fixed amount for each completed chore. He wants to calculate his total earnings based on the number of chores he completes.
6. Anishka is a proactive individual who wants to invest her money wisely. She decides to invest ₹10,000 in a savings account with an annual interest rate of 4%. Anishka is interested in understanding how her investment will grow over time.
7. Aditya is a travel enthusiast who is planning a road trip. He wants to rent a car for his journey. The rental company charges a daily rental fee of ₹300 and an additional cost of ₹2.50 per kilometer driven. Aditya wants to calculate the total cost of renting the car based on the number of days he rents it and the distance he plans to drive.

Solution to above problems

1. Intuition Explanation:
   • For each pot, the cost is ₹100.
   • For each bag of soil, the cost is ₹50.
   • The total cost will be the sum of the costs of pots and bags of soil.

   Mathematical Representation: Using the same variables, let’s represent the number of pots (x) and the number of bags of soil (y), and the total cost (C) can be represented by a linear equation:C=100x+50y

   In this equation:

   • $100x$represents the cost of the pots (since each pot costs ₹100).
   • $50y$represents the cost of the bags of soil (since each bag costs ₹50).
   • The equation adds the two costs to calculate the total cost of the purchase.

   For instance, if Raj wants to buy 3 pots ($x=3$) and 5 bags of soil ($y=5$), Garima can calculate the total cost using the equation:

   $C=100\times 3+50\times 5=300+250=550$
   

   The total cost for Raj’s purchase would be ₹550.

   This example now incorporates how linear equations can be used to model real-world situations in a specific context.

2. Intuition Explanation:
   • The car’s fuel efficiency is 30 miles per gallon.
   • Gasoline costs ₹3.50 per liter.
   • The total cost will be based on the distance Arpit plans to drive and the fuel efficiency of his car.

   Mathematical Representation: Using the same variables, let’s represent the distance (x) Arpit plans to drive and the total cost (C) for the gasoline. The linear equation for the cost can be represented as:

   $C=\frac{x}{30}\times 3.50$ 

   In this equation:

   • $\frac{x}{30}$represents the number of gallons of gasoline needed for the distance $x$(since the car’s fuel efficiency is 30 miles per gallon).
   • $3.50$represents the cost of gasoline per liter in Indian Rupees (INR).
   • The equation calculates the total cost for the gasoline.

   For instance, if Arpit plans to drive 200 miles ($x=200$), the total cost for gasoline can be calculated using the equation:

   $C=\frac{200}{30}\times 3.50=6.67\times 3.50=23.34$
   The total cost for gasoline for this trip would be ₹23.34.

   This example tells how linear equations can be used to model the road trip and gasoline cost scenario.

3. Intuition Explanation:
   • Rohan’s bicycle travels at a constant speed of 15 miles per hour.
   • He wants to calculate the distance he can cover based on the time he rides.

   Mathematical Representation: Using the same variables, let’s represent the time ($t$) Rohan rides and the distance ($d$) he covers. The linear equation for distance can be represented as:

   $d=15t$
   

   In this equation:

   • $15$represents Rohan’s constant speed in miles per hour.
   • $t$represents the time in hours that Rohan rides.
   • The equation calculates the distance he can cover.

   For instance, if Rohan rides for 2.5 hours ($t=2.5$), the distance he can cover can be calculated using the equation:

   $d=15\times 2.5=37.5$
   

   Rohan can cover a distance of 37.5 miles in this time.

   Linear equations provide a way to represent the relationship between these variables and help us calculate distances based on specific time intervals.
4. Intuition Explanation:
   • Syed knows that temperature in Celsius ($C$) can be converted to Fahrenheit ($F$) using the linear equation $F=\frac{9}{5}C+32.$
     
   • He wants to understand how this equation works for temperature conversion.

   Mathematical Representation: Using the same variables, let’s represent the temperature in Celsius ($C$) and the temperature in Fahrenheit ($F$). The linear equation for temperature conversion can be represented as:

   $F=\frac{9}{5}C+32$
   

   In this equation:

   • $\frac{9}{5}$is the conversion factor to convert Celsius to Fahrenheit.
   • $C$represents the temperature in Celsius.
   • $32$is the offset needed for the conversion.
   • The equation calculates the temperature in Fahrenheit.

   For instance, if Syed wants to convert a temperature of 20 degrees Celsius ($C=20$), the temperature in Fahrenheit ($F$) can be calculated using the equation:

   $F=\frac{9}{5}\times 20+32=36+32=68$

   A temperature of 20 degrees Celsius is equivalent to 68 degrees Fahrenheit.

   This example introduces how linear equations can be used to convert temperatures between different scales.

   $$

5. Intuition Explanation:
   • Shubham will earn a fixed amount for each chore he completes.
   • He wants to calculate his total earnings based on the number of chores he does.

   Mathematical Representation: Using the same variables, let’s represent the number of chores ($x$) Shubham completes and his total earnings ($E$). The linear equation for Shubham’s earnings can be represented as:

   $E=10+5x$
   

   In this equation:

   • $\mathrm{10 }$represents the fixed amount Shubham earns even if he doesn’t complete any chores.
   • $\mathrm{5 }$is the amount he earns for each completed chore.
   • $x$represents the number of chores Shubham completes.
   • The equation calculates Shubham’s total earnings.

   For instance, if Shubham completes 7 chores ($x=7$), his total earnings ($E$) can be calculated using the equation:

   $E=10+5\times 7=10+35=45$ 

   Shubham’s total earnings for completing 7 chores would be $45.

   Linear equations provide a way to calculate earnings based on a fixed amount and the number of completed chores, offering an insight into financial calculations.

    

6. Intuition Explanation:
   • Anishka is investing an initial amount of $1000.
   • The investment has an annual interest rate of 4%.
   • She wants to calculate the growth of her investment over a specific number of years. 

   Mathematical Representation: Using the same variables, let’s represent the number of years ($x$) Anishka invests her money and the amount of money ($A$) she will have after the investment period. The linear equation for investment growth can be represented as:

   $A=1000+0.04\times 1000\times x$
   

   In this equation:

   • $1000$represents the initial amount Anishka invests.
   • $\mathrm{0.04 }$represents the decimal form of the 4% annual interest rate.
   • $x$represents the number of years of investment.
   • The equation calculates the total amount of money Anishka will have after the investment period.

   For instance, if Anishka invests her money for 5 years ($x=5$), her total investment amount ($A$) can be calculated using the equation:

   $A=1000+0.04\times 1000\times 5=1000+200=1200$
   

   Anishka will have $1200 after 5 years of investment.

   Linear equations provide a method to project the growth of an investment over time based on an initial amount and an annual interest rate.
7. Intuition Explanation:
   • The rental company charges ₹300 per day for renting the car.
   • Aditya needs to consider the additional cost based on the distance he plans to drive.

    

   Mathematical Representation: Using the same variables, let’s represent the number of days ($d$) Aditya rents the car and the distance ($km$) he plans to drive. The linear equation for the total rental car cost ($C$) can be represented as: 

   $C=300d+2.50\times km$
   

   In this equation:

   • ₹300 represents the daily rental fee for the car.
   • $d$represents the number of days Aditya rents the car.
   • ₹2.50 represents the cost per kilometer driven.
   • $km$represents the distance in kilometers that Aditya plans to drive.
   • The equation calculates the total cost of renting the car.

   For instance, if Aditya plans to rent the car for 4 days ($d=4$) and drive a distance of 250 kilometers ($km=250$), the total rental car cost ($C$) can be calculated using the equation:

   $C=300\times 4+2.50\times 250=1200+625=1825$
   

   The total cost of renting the car for 4 days and driving 250 kilometers would be ₹1825.

   Linear equations provide a way to calculate the total cost of renting a car based on the number of days and the distance driven.

A linear equation is a mathematical equation that represents a straight line when graphed on a coordinate plane. It involves variables, coefficients, and constants, and is usually expressed in the form:

$y=mx+b$

Where:

• $y$is the dependent variable (output),
• $x$is the independent variable (input),
• $m$is the coefficient of $x$, representing the slope of the line (rate of change),
• $b$is the y-intercept, representing the value of $y$when $x$is 0.
  

Data Science Use Case:

In data science, linear equations play a fundamental role in regression analysis. Linear regression is a statistical method that uses linear equations to model the relationship between a dependent variable and one or more independent variables. It’s widely used for tasks like predictive modeling, understanding correlations between variables, and making forecasts based on historical data.

For instance, in predicting house prices, you might use a linear regression model to relate the price ($y$) to various features such as square footage, number of bedrooms, and location ($x$ variables). The linear equation helps you find the best-fit line that minimizes the difference between predicted and actual values.

Some key reasons why linear equations:

1. Problem Solving: Many real-world problems involve relationships where one quantity changes proportionally with another. For example, when you travel at a constant speed, the distance covered is directly proportional to the time traveled. Linear equations provide a way to formalize and solve such problems.
2. Trade and Commerce: Linear equations were particularly important in ancient civilizations for trade and commerce. They enabled merchants and traders to calculate prices, profits, and quantities based on simple proportional relationships.
3. Geometry and Navigation: Linear equations were used in geometry and navigation to describe straight-line paths and distances. This was important for navigation, cartography, and architecture.
4. Physical Sciences: Linear equations were utilized to describe relationships between physical quantities, such as distance, time, and velocity. This was crucial for understanding and predicting motion and other physical phenomena.
5. Advancements in Algebra: The invention of linear equations was intertwined with the development of algebra. As algebraic notation and methods evolved, mathematicians sought ways to solve systems of equations efficiently, leading to the formalization of linear equations and systems.
6. Scientific Progress: Linear equations provided a foundational framework for solving more complex problems. They served as a starting point for developing more advanced mathematical concepts and techniques.
7. Technology and Engineering: As societies advanced and technology became more prevalent, linear equations were used in engineering, architecture, and other fields to design structures, plan projects, and optimize resources.
8. Predictive Modeling: In modern times, linear equations play a crucial role in predictive modeling, data analysis, and machine learning. They allow us to model relationships between variables, make forecasts, and draw insights from data.

One Variable:

In the realm of mathematics, a linear equation featuring a single variable, denoted as $x$, takes the form $ax+b=0$, where $a$is a non-zero constant. The solution to this equation, skillfully derived through algebraic manipulation, is $x=-\frac{b}{a}$.

Two Variables:

When dealing with two variables, often symbolized as $x$and $y$, a linear equation presents itself as $ax+by+c=0$, with the stipulation that both $\mathrm{a }$and $\mathrm{b }$are not simultaneously equal to zero. If $a$and $b$are real numbers, an intriguing property emerges: a multitude of solutions come into play.

Linear Function:

In the arena of calculus, should $b$not equal zero, the equation $ax+by+c=0$transforms into a captivating linear equation with respect to the single variable $y$, valid for all conceivable values of $x$. This imparts a distinctive solution for $y$, a solution intricately rendered as $y=-\frac{a}{b}x-\frac{c}{b}$. Notably, this formulation engenders a function, whose graph materializes as an elegant line characterized by a slope of $-\frac{a}{b}$and a y-intercept of $-\frac{c}{b}$.

Geometric Insights:

The interplay between linear equations and the Cartesian plane yields an insightful geometric interpretation. Every solution pair $(x,y)$of a linear equation $ax+by+c=0$corresponds to the coordinates of a point situated within the Euclidean plane. Intriguingly, these solutions collectively trace out a linear path, unless both $a$and $b$assume the value of zero. Equally captivating is the converse – any straight line can be precisely described by the set of solutions to an appropriately tailored linear equation.

Equation of a Line:

Numerous methods lay at our disposal for defining a line. One such method, the slope-intercept form, leverages a line’s slope $\mathrm{m }$and its y-intercept $y_0$, encapsulated neatly as $y=mx+y_0$. If the line’s inclination is non-horizontal, the equation takes a slightly altered guise, $y=m(x-x_0)$, with $x_0$representing its x-intercept.

Expansions and Symmetry:

The elegance of linear equations lies not only in their definitions but also in their symmetrical expressions. The determinant form, reminiscent of matrices, elegantly captures the essence of the line through a concise arrangement of coefficients. This symmetrical quality allows for intuitive interchange of points without altering the fundamental equation.

Beyond Two Variables:

Delving into the realm of equations featuring more than two variables, we find an adaptable formulation: $a_1{x}_1+a_2{x}_2+\dots +a_n{x}_n+b=0$. Such equations, teeming with coefficients, offer a rich space for exploration and solution. This multitude of variables gives rise to higher-dimensional hyperplanes, adding depth to our understanding.

In this intricate mathematical dance, linear equations stand as pivotal tools, serving as elegant means to express relationships, solve problems, and capture the essence of lines and planes, all while fostering a deeper connection between the abstract and the tangible.