Simplification in equations

When we simplify an equation, it means that we rewrite it into a simpler form by using the calculation rules. We remove parentheses, collect like terms, and make the expressions on both sides of the equals sign more manageable. The simplification does not change the solution, but makes it easier to isolate the unknown.

 

Example 1: Removing parentheses

Solve the equation:

 

$$ \large 2(x+3) = 14 $$

 

We multiply 2 into the parentheses:

 

$$ \large 2x + 6 = 14 $$

 

Now the parentheses have been removed, and the equation is simplified.

 

 

Example 2: Collecting like terms

Solve the equation:

 

$$ \large 3x + 2x - 5 = 20 $$

 

We collect like terms on the left side:

 

$$ \large 5x - 5 = 20 $$

 

The equation is simplified, and we can now isolate \(x\).

 

 

Example 3: Fractions in equations

Solve the equation:

 

$$ \large \tfrac{x}{2} + \tfrac{x}{3} = 10 $$

 

We find a common denominator (6) and rewrite:

 

$$ \large \tfrac{3x}{6} + \tfrac{2x}{6} = 10 $$

 

Combine the fractions:

 

$$ \large \tfrac{5x}{6} = 10 $$

 

The fractions are simplified, and we can now isolate \(x\).

 

 

Example 4: Moving terms

Solve the equation:

 

$$ \large 4x + 7 = 2x + 15 $$

 

We collect all \(x\)-terms on the left side and the constants on the right side:

 

$$ \large 4x - 2x = 15 - 7 $$

 

This simplifies to:

 

$$ \large 2x = 8 $$

 

Now we can easily isolate \(x\).

 

 

Example 5: Combination of steps

Solve the equation:

 

$$ \large 3(x-2) + 5 = 2(x+4) $$

 

First we multiply into the parentheses:

 

$$ \large 3x - 6 + 5 = 2x + 8 $$

 

Simplify the left side:

 

$$ \large 3x - 1 = 2x + 8 $$

 

Move terms:

 

$$ \large 3x - 2x = 8 + 1 $$

 

Simplifies to:

 

$$ \large x = 9 $$

 

 

Summary

When you simplify equations, remember:

 

• Remove parentheses by multiplying in or changing signs.
• Collect like terms on the same side.
• Find a common denominator if there are fractions.
• Move terms between sides so that unknowns are collected.
• Simplify step by step until the equation is ready to solve.

 

Simplification in equations is an important step in solving because it makes the equation clearer and prepares it for isolating the unknown.