Standard Form of Linear Equations

An equation is called linear, or a first-degree equation, when the variable has an exponent of 1. For example, 3x - 2y = 5 is a linear equation in standard form, as x and y both have the highest power of 1.
 

The standard form of a linear equation looks like ax + by = c, where a, b, and c are integers, and x and y are the variables. 

Standard Form of Linear Equations in One Variable

A linear equation with one variable involves only one variable and has only one solution. It is written in its standard form as ax + b = 0. Here, a and b are integers, and x is the variable.
For example, 5x - 10 = 0 has only one solution, x = 2, and is a linear equation in one variable.

Standard Form of Linear Equations in Two Variables

When a linear equation has two variables, it can be written in standard form as: 
ax + by = c
Where: a, b, and c are integers. 
x and y are variables. For example, 2x - 5y = 10 is in standard form and involves two variables.
 

The standard form of linear equations has a wide range of applications in various fields. Some of these applications are mentioned here:

Allocating funds in budgeting and financial planning

Standard form equations can help us plan and track spending between two categories, like rent and groceries.
Example:
Imagine we have a budget of $10,000 for the month. We want to split this between rent (x) and groceries (y).
We can write the equation as:
x + y = 10,000
This shows that whatever we spend on rent plus groceries must add up to $10,000. We can then substitute different values to see how changes in one expense affect the other.

Planning trips using speed and distance problems

While planning trips, we can understand how much speed will be required to cover a given distance and what the travel time will be. For instance, a person wants to travel 300 km; they drive a distance x at 60 km/h, and the remaining distance y at 40km/h. Then, the equation x + y = 300 shows how the total distance is divided between the two speeds. 

Diet and nutrition planning

Dietitians use linear equations to create balanced diets by dividing the nutritional requirements of their clients among different food groups. For example, if 100g of protein needs to be consumed in a day, it can be split between chicken (x) and lentils (y) using the equation x + y = 100.

Calculating capacity in logistics

In transportation and storage, linear equations help figure out the best way to load items without going over weight or space limits.

Example:
A truck can carry up to 1,000 kg. If we load boxes of books (x) that weigh 20 kg each and boxes of tools (y) that weigh 50 kg each, the equation would be:
20x + 50y = 1000
This helps us plan how many of each type of box can fit without overloading the truck.

Seating arrangements in event planning

Event planners use linear equations to figure out how many seats can be set aside for different groups, like VIP and regular guests, while staying within the total capacity.

Example:
Suppose there are 5,000 total seats at a concert. If x represents VIP seats and y represents regular seats, the equation would be:
x + y = 5000
This helps planners decide how many of each type of seat to assign.