**Mastering Cross Multiplication: A Step-by-Step Guide**

Cross multiplication is an essential technique in algebra, used to solve linear equations and inequalities that involve fractions or decimals with variables on both sides of the equation. In this comprehensive guide, we’ll dive into the world of cross multiplication, breaking down each step with clear examples and illustrations.

**What is Cross Multiplication?**

Cross multiplication is a process where you multiply two expressions together while ensuring the variable terms are matched correctly. This technique helps simplify fractions or decimals containing variables by eliminating them and obtaining a solution in its simplest form.

**Why Do We Need Cross Multiplication?**

When working with linear equations involving variables, fractions, or decimals on both sides of an equation, cross multiplication becomes necessary to solve for the unknown variable(s). Without this method, we would struggle to isolate the variable from these expressions. Let’s illustrate this concept:

Suppose you possess an equation like 3x + 2 = (1/2)x – 5

At first glance, solving for x without cross multiplication seems challenging because of the mixed expressions with variables and decimals/fractions. That’s where cross multiplication comes into play!

**Step-by-Step Guide to Cross Multiplication**

Here’s how you can successfully employ cross multiplication:

### Step 1: Ensure Both Expressions Contain Variables

Look at both sides of your equation, making sure that the variable (in this case, x) is present on each side. If one side doesn’t contian a variable term, it may require simplification before proceeding.

Let’s revisit our previous example:

3x + 2 = (1/2)x – 5

We can see that both sides have ‘x’ terms; now we’re ready to begin the cross multiplication process!

### Step 2: Multiply Both Expressions by the LCM of Numerators and Denominators

Calculate the least common multiple (LCM) for any denominators present on each side. For this example:

1/2 = 0.5, which means you’ll multiply both sides by the reciprocal of its denominator, i.e., 2.

(Recall: When multiplying fractions or decimals with variables, ensure you match variable terms correctly.)

So,

- Left-hand side (LHS): Multiply all components involving x and constants by 2.

3x + 2 → (3 × x) × 2 + (1 × 2) × 2

6x + 4

(For decimal expressions with variables, multiply the entire fraction or number on both sides.)

### Step 3: Eliminate Common Terms

Now you can simplify your expression by canceling out any like terms. In this case:

- Remove common variable terms (the ‘6x’ part):
- On LHS: Subtracted from both sides.

6x + 4 = ?

Combine constant parts (numbers with no variables):

+3 and +2

Add/subtract to combine these constants, yielding:

12

Left-hand side is simplified; your goal is to ensure the same on the right-hand side!

### Step 4: Apply Multiplication or Addition/Subtraction as Needed**

Adjust any remaining expression(s) by performing inverse operations. To simplify (1/2)x – 5:

- Multiply both sides by the LCM of numerators and denominators, which was calculated in Step 2: the reciprocal of the original denominator.

So, multiply right-hand side (RHS):

(1 × x)/2 + (-5) × 2

(x)/4 -10

### Step 5: Combine and Simplify**

Add/subtract any constants on both sides to ensure you have equivalent simplified expressions. Your goal is to cancel out variables by combining like terms:

Left-hand side (LHS):

6x + 4

RHS:

(x)/4 -10

Combine constant parts:

+3 -20

Subtract ’10’ and then add ‘+3’:

-7

Final step: Write your final simplified equation. Compare left and right sides, as these should equal each other after cross multiplication:

6x + 4 = (1/2)x – 5

(As expected, the variable terms have canceled out; we’ve achieved equality.)

**Conclusion**

By following this comprehensive guide on how to master cross multiplication, you’ll gain confidence in tackling linear equations with mixed expressions and decimals or fractions containing variables. Remember:

- Ensure both expressions contain variables.
- Multiply both sides by the LCM of numerators and denominators.
- Eliminate common terms through simplification.
- Apply multiplication or addition/subtraction as needed to adjust remaining expression(s).
- Combine, simplify, and write your final equation.

Now you’re equipped to take on any algebraic challenge!