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When you first learn algebra, you tackle simple equations like x = 5 + 4 or y = 5(2 + 1). As you progress, you’ll encounter equations where variables appear on both sides, such as 3x = x + 4 or y² = 9 – 3y². Don’t panic—follow these systematic steps to isolate the variable.

1. Group the Variables on One Side

Move all variable terms to one side, typically the left. For 3x = x + 4, subtract x from both sides: 3x – x = 4. This yields 2x = 4.

TL;DR (Too Long; Didn’t Read)

Adding the additive inverse of a variable to both sides eliminates it from one side.

2. Remove Coefficients

Divide both sides by the coefficient of the variable. From 2x = 4, dividing by 2 gives x = 2.

Another Example

Now consider an equation with an exponent: y² = 9 – 3y².

1. Group the Variables on One Side

Add 3y² to both sides: y² + 3y² = 9. Simplify to 4y² = 9.

2. Remove Coefficients

Divide by 4: y² = 9/4.

3. Solve for the Variable

Take the square root of both sides: y = 3/2.

A Special Case: Factoring

When terms have different degrees, factoring may be required. For example, x² = –2 – 3x.

1. Group the Variables on One Side

Add 3x to both sides: x² + 3x = –2.

2. Prepare for Factoring

Add 2 to both sides to create a zero constant: x² + 3x + 2 = 0.

3. Factor the Polynomial

(x + 1)(x + 2) = 0.

4. Find the Roots

Set each factor to zero: x + 1 = 0 → x = –1; x + 2 = 0 → x = –2. Both satisfy the original equation.