Linear Equation in One Variable

In a linear equation, each variable has a degree of exactly 1.  A linear equation in one variable contains only a variable and results in just one solution.

• When we draw the linear equation, it makes a straight line.
   
• Depending on the equation, the graph can be a slanted, horizontal, or vertical line.
  
  
   
• The general form is ax + b = 0, where x is an unknown variable and a and b are constants. 

For example, adding 7 to an unknown number gives 25. 
In this example, there is only one unknown variable. 
x + 7 = 25 

A linear equation in one variable has only one variable, and does not include squared terms or similar higher powers. The highest degree of such equations is 1.

Solving a linear equation in one variable can be done using the following methods.

1. Balancing Method
2. Transposition Method

• Balancing Method

In the balancing method, the equation is like a weighing scale; both sides must stay equal.

To solve an equation using balance method, we must do the same thing to both sides:

1. Add the same number on both sides.
2. Subtract the same number from both sides
3. Multiply or divide both sides by the same non-zero number to solve for the variable.
4. Move the term to the other side by changing its sign.

Example: x - 3 = 7
Add 3 to both sides to eliminate the -3.
\(x - 3 + 3 = 7 + 3\\ x = 10\)

• Transposition Method

The transposition means moving a term from one side to the other side by changing its sign. 

Example: x + 5 = 12
Move 5 to the other side; it becomes -5.
\(x = 12 - 5\\ x = 7\)
 

Some equations have variables on one side. To solve these, move the number to the other side, and use the opposite operations to isolate the variable.

Let's understand this using few examples for practice.

Example 1: 2x - 4 = 10

Explanation:

1. Add 4 to both sides of the equation,
   \(2x - 4 + 4= 10 + 4\\ 2x = 14\)
    
2. Perform division by 2 on both side.
   \({ 2x \over 2} = {14 \over 2} \\x = 7\)

Example 2: \({2 \over 3} x + {3 \over 6} = 2\)

Explanation:

1. Multiply both side by the LCM, which is 12
   \(12 \times {2 \over 3} x + 12 \times {3 \over 6} = 12 \times 2 \\ 8x + 6 = 24\)
    
2. Perform division by 2 on both side.
   \({ 8x \over 2} + {6 \over 2} = {24 \over 2} \\ 4x + 3 = 12\)
    
3. Move 3 to the other side
   \(4x = 12 -3\\ 4x = 9\)
    
4. Divide both side by 4
   \({4x \over 4} = {9 \over 4} \\ x = {9 \over 4}\)

To understand and effenciently solve linear equations in one variable, here are a few tips and tricks:

1. Always perform arithmetic operations on both side of the equation.
    
2. Simplify the equation first to make calculation easy.
    
3. Don't forget the negative signs.
    
4. If the equations has fractions, eliminate them by multiplying the entire equation with the LCM.
    
5. Remember the sign changes when the number moves to another side.
   
   \(+ \rightarrow - \\ - \rightarrow + \\ \times\rightarrow \div \\ \div\rightarrow \times \)

Parent Tip: Encourage your child to pratice problems from worksheet. Use real life examples to express linear equations to better visualize the linear equations

Linear equations in one variable are useful when only one unknown quantity needs to be found. Here are some real-life applications of linear equations.

1. Finance and Budgeting: It is used to track expenses and income, calculate savings, or planning for future spending. If your income is fixed and expenses vary, a linear equation helps you to solve for what you can afford and how much you are left with.
   
    
2. Shopping and Retail: Retailers use linear equations to find discounts, final prices after applying discounts, offers, or adding taxes, and different charges.
   
    
3. Education and Exams: Linear equations help us determine the required scores, averages, or marks needed to improve grades.
   
    
4. Salaries: They are used to calculate the total pay, including bonus, overtime, and deductions while calculating salaries.