Multi-Step Equations

The algebraic problems that take more than one step to solve are known as multi-step equations. To find the value of the variable, we might need to add, subtract, multiply, or divide. Sometimes, we also have to combine like terms or use the distributive property. 

Inverse operations are mathematical operations that undo each other. For example, addition undoes subtraction, and the inverse of multiplication is division. We use these opposite operations to cancel out terms and solve for the variable in an equation. 
 

To solve multi-step equations, follow these steps:

Step 1 - Simplify both sides of the equation
If the equation has any parentheses, simplify them first. Then, combine any like terms if needed. 

Step 2 - Move all variable terms to one side of the equation
Addition and subtraction can be used to bring all variables to the same side of the equation. When moving terms across the equal sign, change its sign.

Step 3 - Isolate the variable
Use addition or subtraction to move other terms away from the variable. Then, use multiplication or division to get the variable by itself.

Step 4 - Check the solution
Substitute the answer back into the original equation. If both sides are equal, then the solution is correct.

Let’s take an example and apply these steps to find the solution
Example: 3(x - 2) + 4 = 2x + 6

Step 1:
Left side - 3 (x - 2) + 4 = 3x - 6 + 4 = 3x - 2
Right side - 2x + 6 (already in simplest form)
Now, the equation is
3x - 2 = 2x + 6

Step 2: 
Subtract 2x from both sides
3x - 2 - 2x = 2x + 6 - 2x
x - 2 = 6

Step 3:
Separate the variable, add 2 to both sides
x - 2 + 2 = 6 + 2
x = 8

Step 4:
Substitute x = 8 in 3(x - 2) + 4 = 2x + 6
3(8 - 2) + 4 = 2(8) + 6
3(6) + 4 = 16 + 6
18 + 4 = 22
22 = 22
LHS = RHS, so x = 8 is correct.
 

When solving equations that include fractions, it's helpful to eliminate the fractions first. This makes the equation easier to work with. Follow these steps:

Step 1: Find the least common denominator (LCD) for all the fractions in the equation. 

Step 2: Multiply every term on both sides of the equation by that LCD. This clears out the denominators.

Step 3: Once the equation has no fractions, solve it using the usual steps—use inverse operations to isolate the variable.
 

We must understand the following concepts to work with multi-step equations:

1. Variable: A letter or symbol that stands for an unknown number, like x, y, or z.
2. Constant: A number that doesn’t change, like 1.2, 4, -11, etc.
3. Inverse Operations: Pairs of operations that undo each other—like adding and subtracting, or multiplying and dividing.
4. Balance Rule: To keep an equation fair, whatever you do to one side, you must also do to the other.
    

Multi-step equations have many real-life applications and some of them are discussed below:  

Salary and tax calculations
To find someone's net salary, we need to subtract things like tax, insurance, and pension from their gross salary. This kind of problem can be solved using multi-step equations, where each deduction is handled step by step.

Mobile and utility bill calculations
Monthly utility bills include different variables like base fees and usage charges, so they are calculated in the form of multi-step equations.

Profit and cost analysis in business
Businesses use equations to find break-even points or profits by subtracting costs from revenue.

Budgeting for travel
When planning trips, equations can help estimate total costs.

Health tracking in fitness apps
Applications use multi-step equations for calculations like calories burned and consumed periodically.