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 involve performing more than one operation to find the value of a variable. With a clear order of steps and regular practice, solving them becomes quick and accurate.

• Simplify both sides of the equation by combining like terms and removing parentheses.
  
   
• Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable step-by-step.
  
   
• Always perform the same operation on both sides to keep the equation balanced.
  
   
• Check your solution by substituting the value of the variable back into the original equation.
  
   
• Practice solving equations with fractions and negatives to strengthen your problem-solving skills.

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

•  Budget planning: Used to calculate total expenses, savings, and remaining balance after multiple financial steps.
  
   
• Cooking and recipes: Helps in adjusting ingredient quantities when scaling recipes up or down.
  
   
• Travel planning: Used to find total travel time or cost by combining different modes of transport and rates.
  
   
• Business and profit calculations: Helps determine profit after subtracting multiple costs like production, labor, and taxes.
  
   
• Construction and design: Used to calculate material requirements, area, and dimensions involving several measurements.