Solving Equations, Variable On Both Sides

Unit: Linear Equation with one Variable

Chapter: Solving Equations, Variable on both Sides

Reference: – Understanding Equations with Variables on Both Sides, Applying the Properties of Equality, Combining Like Terms for Simplification, Rearranging and Isolating the Variable, Checking and Verifying Solutions, Real-World Applications of Equations with Variables on Both Sides

After studying this chapter, you should be able to understand:

• Understanding Equations with Variables on Both Sides
• Combining Like Terms for Simplification
• Rearranging and Isolating the Variable
• Real-World Applications of Equations with Variables on Both Sides

1. Understanding Equations with Variables on Both Sides
   • Equations can have unknown quantities, called variables, appearing on both sides of the equality sign.
   • These equations require a systematic approach to isolate the variable and determine its value.
   • The goal is to express the equation in a simpler form where the variable is only on one side.
2. Applying the Properties of Equality
   • The equality of an equation must be maintained throughout the solving process.
   • The same mathematical operation, whether addition, subtraction, multiplication, or division, must be applied to both sides.
   • This ensures that the balance of the equation is preserved and that the solution remains valid.
3. Combining Like Terms for Simplification
   • Terms that contain the same variable must be grouped together to reduce complexity.
   • Constant terms (without variables) should also be combined separately to simplify calculations.
   • This step helps in systematically eliminating unnecessary terms and progressing toward an isolated variable.
4. Rearranging and Isolating the Variable
   • Moving variable terms to one side of the equation and constant terms to the other side is necessary.
   • Using inverse operations allows for the elimination of unwanted terms, helping to isolate the variable.
   • This step is critical for solving the equation and finding a clear value for the unknown.
5. Checking and Verifying Solutions
   • Once a solution is obtained, it must be substituted back into the original equation to confirm its correctness.
   • If both sides of the equation result in the same value, the solution is valid.
   • This step prevents errors and ensures that the solution satisfies the given equation.
6. Real-World Applications of Equations with Variables on Both Sides
   • Such equations frequently appear in real-life situations, such as budgeting, distance-speed-time problems, and scientific calculations.
   • Understanding their applications enhances problem-solving skills and logical reasoning.
   • Mastering these equations helps in fields like finance, engineering, and physics, where precise calculations are essential.

Example: –

A company offers two different mobile data plans:

• Plan A charges a fixed monthly fee of $20 plus $2 per GB of data used.
• Plan B has no fixed fee but charges $4 per GB of data used.

A customer wants to determine after how many GB of data usage the cost of both plans will be the same.

Using equations with variables on both sides, simplification techniques, and verification, find:

1. The number of GB at which both plans cost the same.
2. Verify the solution by substituting it back into the equation.

Solution: –
 

Step 1: Forming the Equation

Let x be the number of GB of data used.

• Cost of Plan A = Fixed fee + (Cost per GB × Data usage)

20+2x

• Cost of Plan B = Cost per GB × Data usage

4x

Since we are looking for the point where both plans cost the same:

20+2x=4x

Step 2: Solving the Equation

Rearrange and isolate the variable by moving all terms with x to one side:

20=4x−2x

Divide both sides by 2:

x=20/2 = 10

Thus, the data usage at which both plans cost the same is 10 GB.

Step 3: Verifying the Solution

Substituting x = 10 back into both cost equations:

• Plan A cost

20+2(10) =20+20=40

• Plan B cost

4(10) =40

Since both costs are equal ($40), our solution is verified.

Conclusive Points for "Solving Equations with Variables on Both Sides"

1. Equations with variables on both sides require systematic simplification
   • To solve such equations, the variable terms must be brought to one side, while constant terms are moved to the other.
2. Maintaining equality is crucial in every step
   • Applying the same mathematical operations to both sides ensure the balance of the equation remains unchanged.
3. Combining like terms simplifies the equation
   • Grouping similar terms helps in reducing complexity and making the equation easier to solve.
4. Checking the solution ensures accuracy
   • Substituting the obtained value back into the original equation helps verify correctness and avoid mistakes.
5. Such equations have practical applications in real-world problem-solving
   • These equations are used in business, physics, finance, and engineering, making them an essential mathematical concept.